LIBRAW 


BEVERLEY  EDUCATIONAL  SERIES 

EDITED  BY 

W.  W.   CHARTERS 

PROFESSOR    OF    EDUCATION 
CARNEGIE   INSTITUTE   OF  TECHNOLOGY 


SCHOOL  STATISTICS  AND 
PUBLICITY 


BY 

CARTER  ALEXANDER 

FIRST  ASSISTANT  STATE  SUPERINTENDENT  OF  PUBLIC 

INSTRUCTION  FOR  WISCONSIN 

SOMETIME   PROFESSOR   OF   SCHOOL  ADMINISTRATION 

GEORGE   PEABODY   COLLEGE   FOR   TEACHERS 


SILVER,   BURDETT  AND   COMPANY 

BOSTON  NEW  YORK  CHICAGO 


Copyright,  1910,  by 
CAKTEIi   ALEXANDER 

All  rights  reserved 


EDITOR'S  PREFACE 

We  hear  from  the  forum  and  pulpit  that  reconstruction 
must  follow  a  war  of  such  magnitude  as  that  just  closing. 
But  the  laws  of  habit  still  operate  and,  if  permitted  to  do 
so,  the  nation  will  return,  in  fact  will  prefer  to  return,  to 
those  accustomed  grooves  of  thought  and  action  from 
which  they  have  been  so  vigorously  shaken. 

This  return  is  not  possible  in  those  cases  which  have  an 
economic  basis.  The  industrial  world  will  be  compelled 
by  the  insistence  of  labor  to  change  its  ideals  and  practices, 
and  problems  of  finance,  taxes,  and  revenues  will  force 
reconstruction  in  the  field  of  politics.  But  education  will 
not  feel  the  insistent  urge  of  economic  forces  unless  some 
obvious  catastrophe,  such  as  the  spectacle  of  schoolrooms 
unprovided  with  teachers  because  of  public  parsimony, 
visualizes  the  crisis  for  the  taxpayers.  Situations  less 
obvious  than  this  the  citizen  does  not  see  and  understand. 
It  is  difficult  to  show  him  that  low  salaries  may  result  in 
other  evils  as  serious  as  the  corporeal  absence  of  teachers 
from  the  classroom.  He  must  be  shown  that  not  only 
must  teaching  be  good  but  that  better  buildings,  better 
sanitation,  curriculum  changes,  compulsory  education, 
and  vocational  education  must  be  provided. 

Sonorous  or  staccato  generalities  about  reconstruction 
which,  I  venture  to  say,  in  less  than  a  twelvemonth  of 
our  swift-moving  current  of  national  life  will  have  be- 
come platitudinous,  will  not  produce  that  sympathetic 
attitude  toward  educational  reconstruction  which  is  neces- 


vi  Editor's  Preface 

sary  for  its  processes  to  be  carried  on.  Antebellum  habits 
will  wage  a  momentous  warfare  upon  transitory  emotional 
ideals. 

And,  even  though  the  present  moment  should  be  preg- 
nant with  sympathy  for  the  improvement  of  the  schools, 
other  obligations  compete  for  a  place  in  the  necessarily 
limited  field  of  public  attention.  National  taxes  com- 
pete with  school  taxes,  political  conflicts  are  more  promi- 
nent than  school  necessities,  and  industrial  readjustments 
obtain  readier  attention  than  educational  reconstruction. 

To  meet  this  situation  the  man  of  the  hour  is  the  super- 
intendent of  schools.  The  problem  of  crystallizing  the 
liquid  desires  of  the  public  for  educational  progress  is  as 
squarely  placed  upon  his  shoulders  as  any  human  task 
has  ever  rested  upon  any  individual.  For  education  is 
personal.  The  nation  is  educated  in  community  groups. 
Just  as  the  squad  is  the  unit  of  the  army,  the  community 
is  the  unit  of  the  nation,  and  just  as  the  non-commissioned 
officer  who  leads  the  squad  is  the  "backbone  of  the  army," 
so  the  superintendent  who  leads  his  community  is  the 
backbone  of  the  forces  of  education.  The  educational 
army  is  just  as  dependent  upon  his  intelligence  and  in- 
dustry as  the  fighting  army  is  dependent  upon  the  cor- 
poral. 

The  chief  weapon  for  leading  the  people  of  a  community 
in  educational  activity  is  publicity.  And  this  may  be 
obtained  in  three  ways.  One  method  is  to  develop  ex- 
cellent schools  and  let  the  work  speak  for  itself  through 
satisfied  parents,  loyal  teachers,  and  efficient  children. 
But  this  method  as  the  sole  method  is  open  to  the  criti- 
cism that  parents  and  children  cannot  clearly  distinguish 
between  good  school  systems  and  inferior  systems,  and 
to  the  further  criticism  that  if  people  are  too  well  satisfied 


Editor's  Preface  vii 

with  their  schools  they  are  not  sensitive  to  the  need  for 
better  schools.  So,  important  as  the  maintenance  of  a 
good  system  is,  its  presence  does  not  insure  knowledge  of 
its  excellence  and  needs. 

A  second  method  is  personal  explanation  of  what  the 
school  is  doing  and  attempting,  carried  on  by  conferences 
with  the  school  board,  by  public  meetings,  and  by  private 
conversation.  But  a  third  method  to  be  added  to  these 
is  the  superintendent's  annual  report  and  printed  com- 
munications, in  which  a  wider  audience  is  reached.  All 
three  of  these  methods  are  used  by  those  superintendents 
who  have  been  most  successful  in  molding  public  opinion. 

But,  as  Mr.  Alexander  points  out,  seventy  per  cent  of 
a  group  containing  one  hundred  twenty-eight  members 
of  two  of  the  most  distinguished  organizations  of  an  in- 
telligent community  stated  that  they  did  not  read  the 
school  board  reports.  And  this  proportion  is  small  for 
the  nation  as  a  whole. 

Upon  the  problem  of  making  the  superintendent's 
report  readable  by  his  community  this  text  is  directed. 
The  author  attacks  the  whole  problem  from  the  collect- 
ing of  the  data  and  their  statistical  treatment,  to  the 
presentation  of  his  findings  in  simple  and  graphic  form. 
It  is,  therefore,  presented  as  a  notable  attempt  to  make 
known  to  the  public  those  inner  workings  of  the  school, 
to  the  end  that  fluid  educational  interest  may  take  on 
stability  of  action  directed  toward  progressive  ends. 


AUTHOR'S  PREFACE 

The  admitted  ineffectiveness  of  our  school  statistics  is 
due  mainly  to  two  causes.  First,  many  superintendents 
do  not  know  how  to  apply  statistics  as  well  as  do  their 
equals  in  intelligence  in  other  fields.  Second,  these  super- 
intendents do  not  -  know  —  because  they  have  been  too 
busy  to  learn  —  how  to  present  statistical  matter  to  the 
public  effectively. 

As  the  experience  of  publicists  in  other  fields  shows,  the 
ways  of  doing  these  things  are  simple.  But  not  enough 
advantage  of  the  labors  of  these  men  has  been  taken,  nor 
have  their  results  been  adapted  for  easy  and  quick  use 
by  the  busy  superintendent.  There  is  pressing  need  for 
a  book  which  will  do  these  very  things.  The  aim  of  this 
book,  then,  is  to  make  available  for  superintendents  and 
classes  in  school  administration  the  results  of  years  of 
study  of  statistical  theory  and  its  applications  to  school 
data  for  publicity  purposes,  as  shown  in  school  reports 
and  surveys. 

These  results  have  all  passed  through  the  fires  of  criti- 
cism of  the  large  number  of  practical  school  men  whom 
it  has  been  the  author's  good  fortune  to  have  as  students 
during  the  past  six  years.  In  particular,  he  wishes  to 
acknowledge  his  great  indebtedness  to  the  men  of  Ed- 
ucation 245,  his  graduate  course  at  Peabody,  whose 
generous  aid  and  criticism  have  contributed  much  to 
the  practical  helpfulness  of  the  book. 

Acknowledgments  are  also  gratefully  made  to  the  fol- 


x  Author's  Preface 

lowing:  Professor  E.  L.  Thorndike  of  Teachers  College, 
who  introduced  the  writer  to  statistical  method ;  the 
editor,  Professor  W.  W.  Charters,  who  has  given  valuable 
suggestions  for  modifying  the  form  of  the  book ;  Mr. 
W.  C.  Brinton  and  Professors  J.  F.  Bobbitt  and  W.  I. 
King,  on  whose  writings  the  author  has  drawn  freely ; 
Dr.  W.  W.  Theisen,  supervisor  of  educational  measure- 
ments in  the  Wisconsin  State  Department  of  Public  In- 
struction, who  has  read  part  of  the  proof ;  the  various 
writers  and  foundations  who  have  given  permission  for 
the  reproduction  of  copyrighted  charts;  Messrs.  C.  H. 
Moore,  E.  McK.  Highsmith,  and  S.  C.  Garrison  for  careful 
criticism  of  the  manuscript. 

A  special  feature  of  this  book  is  that  all  but  five  of  the 
cuts  are  from  drawings  made  by  the  boys  in  the  Tech- 
nological High  School  at  Atlanta  under  the  direction  of 
Mr.  E.  S.  Maclin,  at  that  time  head  of  the  drawing  de- 
partment there.  Mr.  Maclin,  who  was  one  of  the  men 
in  Education  245,  has  kindly  given  his  services  to  produce 
this  practical  demonstration  of  the  possibilities  in  utilizing 
high  school  students  on  school  publicity.  No  doubt  there 
are  some  slight  errors  or  irregularities  to  be  found  in  the 
cuts.  But  all  such  imperfections  are  only  evidences  of 
the  genuineness  of  the  demonstration. 

The  school  superintendent  must  be  a  publicist.  He 
must  make  reports  to  the  public.  In  many  places  for 
the  next  decade  at  least  he  must  fight  as  hard  as  any  officer 
in  the  trenches  to  ward  off  the  incessant  and  fierce  attacks 
made  upon  his  school  appropriations  by  politicians  and 
hard-pressed  but  unthinking  tax-payers.  For  warding 
off  or  beating  back  such  attacks,  his  most  effective  weapons 
will  be  reports  containing  simple  but  skillful  statistical 
devices  for  presenting  the  claims  of  the  school   children. 


Author's  Preface  xi 

Unless  he  has  such  weapons,  the  enemy  will  be  liable  to 
sweep  over  the  schools  and  place  them  on  a  starvation 
diet. 

If  this  book  in  any  material  way  helps  the  superin- 
tendent of  schools,  active  or  in  training,  to  arouse  his 
public  and  to  secure  adequate  support  for  his  school,  it 
will  fulfill  its  mission. 

Carter  Alexander. 


SUGGESTIONS   FOR  USING  THE   BOOK 

I.   TO   THE   SUPERINTENDENT  IN   THE  FIELD 

The  active  superintendent,  accustomed  as  he  is  to 
adapting  materials  quickly  to  his  own  problems,  of  course 
needs  few  suggestions  for  using  this  book.  But  to  save 
his  time,  a  few  are  given. 

1.  All  of  this  book  except  Chapters  V  to  VII  can 
be  read  with  little  effort  by  any  active  scnool  man.  Ap- 
plications to  his  own  work  will  constantly  come  to  mind. 

2.  A  beginner  in  statistical  method  can  attain  a  fair 
acquaintance  with  the  matters  involved  in  Chapters  V 
to  VII  by  a  careful  reading  of  the  text.  But  a  usable 
knowledge  of  these  chapters,  for  one  not  previously  familiar 
with  statistical  method,  can  hardly  be  expected  unless  the 
exercises  or  similar  ones  are  done  as  indicated  in  the 
suggestions  for  the  instructor  of  future  superintendents. 
For  such  exercises,  the  superintendent  generally  has  on 
hand  several  administrative  problems  that  would  be 
greatly  simplified  by  adequate  statistical  treatment. 
His  annual  report,  in  particular,  will  furnish  material  for 
practically  every  exercise  in  the  book. 

3.  The  table  of  contents  and  the  index  will  enable  him 
to  use  the  book  as  a  reference  book  for  securing  material 
or  suggestions  for  procedure  on  school  problems  necessitat- 
ing statistical  treatment.  He  can  also  quickly  get  ideas 
and  devices  for  presenting  his  results  to  the  public. 

4.  If  he  has  plenty  of  time,  the  procedure  advocated 
for  students  in  training  will  be  very  profitable. 


xiv  Suggestions  for  Using  the  Book 

II.   TO   THE  INSTRUCTOR  ENGAGED  IN   TRAINING  FUTURE 
SUPERINTENDENTS 

The  experience  of  the  writer  with  the  active  superin- 
tendents and  future  superintendents  in  his  own  classes 
indicates  the  following : 

1.  It  is  advisable  to  have  a  good  supply  of  supple- 
mentary material  for  practice  work  and  concrete  illus- 
trations. At  least  one  copy  of  each  reference  given  i,i 
the  bibliography  in  the  back  of  the  book  is  advisable. 
In  addition,  one  should  secure  before  the  class  starts  a 
fairly  complete  collection  of  typical  annual  reports  and 
publicity  pamphlets  of  the  city  and  county  superintend- 
ents in  the  district,  state,  and  section  of  the  country  from 
which  the  students  come.  It  is  well  to  have  several 
copies  of  the  better  ones  of  these,  but  for  most  of  them 
one  copy  each  is  sufficient.  These  should  be  distributed 
among  the  members  of  the  class,  each  being  allowed  to 
choose  those  he  especially  cares  for  and  each  being  re- 
sponsible for  his  own  set.  All  students  may  be  required 
to  become  familiar  with  the  better  ones  for  which  dupli- 
cate copies  are  available. 

2.  Portions  of  the  text  may  be  prepared  for  class  dis- 
cussion as  in  any  textbook  course.  But  the  instructor 
may  test  the  extent  to  which  the  students  have  mastered 
the  text  and  create  much  more  interest  by  constantly 
expecting  them  to  bring  in  pertinent  illustrations  of  the 
points  under  discussion.  These  illustrations  should  be 
obtained  from  the  reports  or  other  material  for  which 
the  student  is  responsible,  from  his  previous  experience, 
or  from  current  magazines  and  advertisements. 

3.  A  mastery  of  the  text,  however,  can  be  assured  only 
when  students  prove  their  ability  to  handle  the  exercises 


Suggestions  for  Using  the  Book  xv 

in  this  book,  or  similar  ones  which  the  instructor  may 
easily  make  up  or  select  from  the  references  in  the  bibliog- 
raphy. "To  know  whether  any  one  has  a  given  mental 
state,  see  if  he  can  use  it." 

Answers  to  exercises  have  been  purposely  omitted. 
Most  students  come  to  such  exercises  with  a  habit  of 
striving  for  an  accuracy  to  several  decimals  in  the  answer 
rather  than  of  concentrating  their  attention  on  the  prin- 
ciples involved.  The  particular  decimals  secured  in  any 
answer  will  often  vary  considerably  according  to  the 
grouping  used,  the  extent  to  which  decimals  are  carried 
out,  and  so  forth.  Again,  many  administrative  problems 
involve  original  data  based  on  approximate  measures, 
where  attempts  to  secure  absolute  numerical  accuracy 
are  a  sheer  waste  of  time  for  everybody.  By  checking 
the  procedure  and  answers  of  various  students  against 
each  other,  a  sufficiently  accurate  answer  will  be  ob- 
tained in  the  class.  To  secure  this  result,  however,  the 
instructor  should  see  that  the  exercises  are  worked  out 
in  full  and  handed  in.  Judging  by  the  writer's  experi- 
ence, it  is  hardly  worth  while  to  discuss  any  of  the  ex- 
ercises that  involve  computation  until  they  have  been 
completely  written  out  by  the  students.  The  work  of 
both  student  and  instructor  will  be  greatly  reduced  for 
exercises  which  occur  on  pages  38  to  316,  by  utilizing  the 
work  already  done  on  a  problem,  merely  adding  the  work 
for  the  particular  point  to  be  illustrated  and  handing  in 
the  exercise  again. 

If  there  is  time  and  the  students  are  sufficiently  mature, 
the  problem  started  in  the  exercises  on  pages  38,  43,  57, 
61,  81,  89  may  be  continued  throughout  the  course  by 
each  student  with  excellent  results.  It  is  not  continued 
in  this  book  because  of  the  difficulty  of  keeping  a  group 


xvi  Suggestions  for  Using  the  Book 

together  on  such  work  in  the  usual  time  allotted  to  reci- 
tations. But  in  the  writer's  own  classes,  his  students 
continue  to  work  outside  on  such  problems  and  make 
oral  or  written  reports  at  the  end  of  the  course.  Prob- 
lems of  this  nature  interest  students  greatly.  Any  in- 
structor can  find  many  opportunities  for  them  in  the 
local  city  school  system,  with  resulting  profit  to  the 
system. 

4.  For  the  convenience  of  the  instructor  and  a  few  of 
the  students  who  evince  unusual  interest  or  capacity  for 
the  work,  a  few  references  are  given  at  the  end  of  various 
chapters.  It  is  not,  however,  advisable  to  use  many  of 
these  for  class  reading.  In  attempting  to  understand 
statistical  method,  the  student  must,  above  everything 
else,  avoid  confusion  and  keep  his  feeling  of  mastery  over 
what  he  has  done.  But  he  will  find  the  various  writers 
taking  up  topics  in  different  sequence  and  using  different 
definitions  or  different  technical  terms  in  a  manner  very 
confusing  to  a  beginner.  Much  better  results  will  be 
secured  if  the  instructor  will  use  most  of  the  references 
himself  and  adapt  the  additional  material  he  wishes  to 
the  plan  of  this  book  and  the  special  needs  of  his  students. 

Experiments  with  the  writer's  own  classes  show  that  a 
satisfactory  working  knowledge  of  thei  material  presented 
in  this  book  can  be  comfortably  acquired  in  from  twenty- 
five  to  forty-five  class  periods.  The  shorter  time  will,  of 
course,  reduce  the  use  of  exercises.  The  longer  time  will 
allow  abundant  opportunity  for  the  use  of  the  exercises 
and  special  problems  advocated.  The  book  can  thus  be 
used  in  a  six  weeks'  summer  school,  in  a  corresponding 
period  during  the  regular  session,  dT  in  whatever  longer 
period  is  available. 


TABLE   OF  CONTENTS 

PAGE 

Chapter  One.    Why  We  Need  Better  School  Statistics        1 
I.     Common  Errors 
II.     Wastefulness  of  These  Errors 

III.  Indifference  of  the  Public 

IV.  Progress  Elsewhere  in  Using  Statistics 

V.     Unsatisfactory  State  of  Affairs  in  School  Statistics 

Chapter  Two.     Collection  of  Data 33 

I.     When  to  Use  Statistical  Method 
II.     How  to  Plan  Statistical  Treatment  of  Problems 

III.  How  to  Determine  Units  and  Scales 

IV.  How  to  Do  the  Actual  Collecting 

Chapter  Three.     Technical  Methods  Needed  in  School 

Statistics 90 

I.     Usual  Views 
II.     Statistical  Knowledge  Needed  for  School  Surveys 

III.  Statistical   Knowledge  Needed   for   Reading   Educa- 

tional Investigations 

IV.  Illustration   of  Value   of   Statistical   Method  to   the 

Superintendent 
V.     Statistical  Method  as  a  Form  of  Expression 

Chapter  Four.    Scales,   Distribution  Tables,  and  Sur- 
faces of  Frequency 100 

I.     Scales 

II.     Distribution  Tables 
III.     Surfaces  of  Frequency 

Chapter  Five.     Measures  of  Type 124 

I.     The  Mode 
II.     The  Median 

III.  The  Average 

IV.  Which   Measure  of  Type  to  Use    in    a    Given    Dis- 

tribution 

xvii 


xviii  Table  of  Contents 

PAGE 

Chapter  Six.     Measures  of  Deviation  or  Dispersion      .     149 
I.     Extreme  Range  Variation 
II.     Quartile  Deviation 

III.  Other  Percentile  Deviations 

IV.  Median  Deviation 
V.     Average  Deviation 

VI.     Standard  Deviation 
VII.     Deviations  for  Skew  Distributions 
VIII.     Which    Measure   of    Deviation   to   Use   in    a    Given 
Distribution  . 

Chapter  Seven.     Measures  of  Relationships    .        .  164 

I.     Relationships  Inside  of  One  Group 
II.     Simple  Relationships  between  Different  Distributions 

III.  Coefficient  of  Variability  or  Dispersion 

IV.  Correlation 

Chapter  Eight.     Supplement  on  Statistical  Treatment  .     187 
I.     Reliability  of  Statistical  Results 
II.     Special  Economies  in  Calculation 
III.     Combining  Data  Given  in  Rank  Order  Only 

Chapter   Nine.    Uselessness   of   Statistics   in   Current 

School  Reports 200 

I.     The  Situation 
II.     Causes  of  the  Uselessness 
III.     Devices  for  Effective  Presentation 

Chapter  Ten.     Presenting  Tabulated  Statistics  to  the 

Public 206 

I.     Possibilities  of  Using  Tabulations  of  Statistics  to  In- 
fluence the  Public 
II.     How  to  Give  a  Bird's-Eye  View  through  Tabulation 
III.     How  to  Make  Up  a  Series  of  Tables  of  the  Same 
General  Nature 

Chapter    Eleven.     Graphic    Presentations    of    School 

Statistics  Especially  for  the  Public       .         .     234 
I.     Object  of  Graphic  Presentations 
II.      How  to  Make  Graphs  for  the  Public  from  Statistical 
Data 


Table  of  Contents  xix 

PAGE 

III.  How  Graphs  for  the  Public  Differ  from  Those  for 

the  Administrator 

IV.  Examples  of  Good  Graphs  on  School  Statistics  for 

the  Public 
V.     Economies  in  Making  School  Graphs  for  the  Public 

Chapter    Twelve.     Translating    Statistical    Material 

on  Schools  for  the  Public         ....     303 
I.     The  Need  of  Translation 
II.     Suggestions  for  Good  Translations 
III.     Examples    of    Good     Translations    of    School    Sta- 
tistics 

Selected  and  Annotated  Bibliography       ....     317 
Index 323 


SCHOOL  STATISTICS  AND  PUBLICITY 

CHAPTER   I 

WHY   WE   NEED   BETTER   SCHOOL   STATISTICS 

I.    COMMON  ERRORS 

Many  school  superintendents  have  no  doubt  laughed 
over  some  variation  of  the  following : 

A  clerk  in  a  department  store  asked  for  a  raise  in  salary  and  got 
this  reply  from  the  owner : 

"Why,  this  is  no  time  to  ask  for  more  wages.  Times  are  too  hard 
and  you  do  very  little  work. 

"I  will  show  you  how  little  work  you  do  for  me  in  a  year. 

"The  year  has  365  days  in  it,  and  each  day  is  24  hours,  divided 
in  three  equal  parts,  8  hours  for  work,  8  hours  for  sleep,  and  8  hours 
for  play. 

"Now  just  listen.  Take  8  hours  that  you  sleep  in  each  day, 
which  is  122  days,  from  365,  and  that  leaves  243  days. 

"You  play  8  hours  each  day,  which  is  another  122  days  from  243, 
and  that  leaves  121  days,  —  see? 

"Now,  there  are  52  Sundays  when  you  do  not  work;  just  take 
these  from  121,  and  that  leaves  69  days. 

"When  the  summer  comes,  you  say:  'I  can't  work,  I'm  all  in, 
and  I  want  a  vacation.'  I  give  you  2  weeks  off;  take  14  days 
from  69.     This  leaves  55  days. 

"Then  the  store  closes  a  half  day  every  Saturday.  This  is  26 
from  55,  which  leaves  29  days. 

"Then  you  take  1|  hours  for  your  lunch  each  day,  which  is  28 
days,  from  29,  or  1  day  left. 

"And  I  just  remember  that  that  day  was  the  retailmen's  picnic, 
and  you  asked  off  to  go  to  it." 

1 


2  School  Statistics  and  Publicity 

The  isolated  statements  seem  fair  enough  on  the  sur- 
face when  taken  one  at  a  time.  What  has  been  done 
to  manipulate  the  figures  so  that  a  patently  erroneous 
conclusion  is  reached  ?  Humorous  as  this  is,  it  is  not  so 
far  beyond  what  can  be  found  sometimes  in  the  statistical 
parts  of  school  reports. 

Let  us  examine  samples  of  these  errors,  which  for  con- 
venience may  be  regarded  as  arising  from  troubles  with : 

1.  Unanalyzed  Totals. 

2.  Comparisons  Employing  Indefinite  Units. 

3.  Comparisons  Using  Unsound  Elementary  Treatment. 

4.  Attempts  at  Too  Great  Accuracy. 

5.  Neglect  of  Technical  Statistics. 

6.  Presentations  of  School  Statistics  to  the  Public. 

These  will  now  be  discussed  in  order,  but  only  for  the 
purpose  of  showing  the  errors  clearly.  The  remedies 
for  such  troubles  will  be  given  in  subsequent  chapters.1 

1.   Unanalyzed  Totals 

Some  commercial  clubs  and  school  systems  print  figures 
on  their  letterheads  giving  the  size  of  the  schools,  number 
of  teachers,  number  of  children  enrolled,  number  of  build- 
ings, size  of  plant,  etc.  A  certain  normal  school  in  a 
middle-western  state  uses  this  material  on  its  letterhead : 

A  Teacher's  College 

Faculty,  46  —  men,  19  ;  women,  27 

Enrollment,  1915-16,  2179 

Average  attendance,  874 

1  For  those  who  wish  to  use  this  book  for  reference  purposes  or 
for  review,  cross  references  to  later  portions  are^iven  throughout 
this  chapter.  But  for  those  who  intend  to  read  the  whole  book  and 
who  are  going  through  the  chapter  for  the  first  time,  it:  is  advisable 
to  ignore  the  cross  referents  at  this  stage. 


Why  We  Need  Better  School  Statistics      3 

These  figures  mean  very  little  unless  the  reader  knows  in 
addition  whether  the  proportion  of  the  number  of  the 
faculty  to  the  number  in  the  student  body  is  somewhere 
close  to  that  of  the  average  normal  school  in  the  country  ; 
whether,  in  averaging  the  attendance,  there  was  one 
short  term  with  a  heavy  attendance  set  over  against 
three  long  terms  with  a  rather  lean  attendance,  etc. 

Another  illustration  of  this  error  is  the  statement 
sometimes  made  when  medical  inspection  is  installed 
in  a  city,  that  the  results  show  90  per  cent  or  over  of  the 
children  to  be  defective.  This  tends  to  alarm  people 
until  they  consider  that  there  must  be  some  fallacy  in 
it  or  there  could  be  no  such  thing  as  normal  children  in 
that  city.  The  fact  is  that  practically  every  child  is 
physically  defective  on  some  point.  When  children  are 
measured  on  ten  to  twenty  physical  points,  each  child 
is  certain  to  be  defective  on  some  one  point,  but  many 
of  them  are,  on  the  whole,  average  or  normally  healthy 
children.  The  unanalyzed  total  gives  a  misleading  im- 
pression. 

2.   Comparisons  Employing  Indefinite  Units 

Even  if  the  error  of  unanalyzed  totals  is  avoided,  a 
second  weakness  may  creep  in  through  foolish  or  mis- 
leading comparisons.  These  may  be  of  this  nature 
because  of  lack  of  suitable  units  for  comparing  the  things, 
or  the  trouble  may  arise  because  of  unsound  methods 
in  making  the  comparisons. 

Enrollment.  For  example,  a  certain  college  at  one 
summer  session  announced  an  enrollment  of  1108.  At 
the  second  summer  session  after  this  the  figures  were 
1484.  Apparently  this  was  a  gain  of  only  34  per  cent. 
In  reality  the  school  had  gained  about  60  per  cent.     The 


4  School  Statistics  and  Publicity 

enrollment  the  first  time  was  padded  by  the  issuing  of 
visitors'  cards,  even  to  wives  of  the  faculty.  The  enroll- 
ment the  third  time  included  only  bona  fide  students, 
owing  to  the  need  of  impressing  certain  people  with  the 
school's  actual  enrollment.  The  difference  in  units  here 
prevented  the  school  from  gaining  the  advertising  value 
of  tha  really  large  increase  of  the  second  year. 

Serious  errors  often  occur  in  comparisons  between 
school  systems  because  the  number  of  children  has  not 
been  figured  on  the  same  basis.  Thus,  this  number  may 
be  calculated  as  the  total  enumerated  of  school  age  (school 
census),  as  the  total  ever  on  the  roll  (enrollment),  as  the 
total  number  in  attendance  that  have  not  been  absent 
over  three  days  (number  belonging),  or  as  the  average 
daily  attendance.  Also,  the  percentage  of  daily  attendance 
is  often  figured  on  the  "  number  belonging  "  base.  This  of 
course  always  gives  a  high  percentage  of  daily  attendance, 
since  absent  pupils  are  rapidly  dropped  from  the  count. 
It  is  in  reality  a  measure  of  the  holding  power  of  the 
schools  for  truants,  rather  than  for  all  children  of  school 
age.  Sometimes  the  enrollment  in  a  school  system  equals 
or  exceeds  the  enumeration,  not  because  of  the  attraction 
for  enumerated  children,  but  because  of  the  enrollment 
of  non-residents,  no  mention  of  which  may  be  made  in 
that  connection.  Or  the  enrollment 'may  be  greater  be- 
cause of  transfers  inside  the  system  so  that  the  same 
pupil  is  enrolled  in  more  than  one  ward  school  and  con- 
sequently is  counted  twice  or  more  on  enrollment. 

School  Expenditures.  Erroneous  conclusions  are  often 
reached  in  attempting  to  compare  the  total  expenditures 
of  one  city  school  system  with  those  oT  another  city, 
without  proper  units.  These  totals  are  vitally  affected 
by  such  factors  as  the  number  of  people  residing  in  the 


Why  We  Need  Better  School  Statistics       5 

city,  the  per  capita  wealth,  the  proportion  of  children 
in  the  population,  the  rate  of  taxation  for  all  purposes, 
etc.  A  rich  suburban  city  can  be  expected  to  expend! 
far  more  on  its  schools  than  can  a  city  of  the  same  size 
made  up  mainly  of  factory  workers  or  similar  wage 
earners. 

Distribution  of  School  Moneys.  In  some  places  in 
the  South,  state  school  money  is  distributed  on  one  unit, 
the  enumeration  basis,  and  expended  on  another,  the 
number  of  children  (more  likely  of  white  children)  in  at- 
tendance. For  example,  some  years  ago  in  Alabama  one 
community  is  said  to  have  had  enough  in  this  way  to  pay 
the  principal  of  the  white  school  $1200  and  give  him 
three  assistants,  all  for  an  enrollment  of  forty-five  white 
children  and  an  average  daily  attendance  of  not  more 
than  thirty-five.  This  was  of  course  achieved  by  spend- 
ing on  the  white  schools  considerable  money  drawn  on 
negro  census  children.1 

Age  of  Child.  Another  indefinite  unit  is  the  "  age  " 
of  a  child  in  any  given  school  system.  Is  a  six-year-old 
child  one  that  is  six  and  has  not  reached  seven,  or  is  it  a 
child  from  five  and  a  half  up  to  six  and  a  half  years  old  ? 
In  making  the  recent  survey  of  Cleveland,  Dr.  Ayres 
found  that  the  age  of  a  child  in  that  city  was  determined 
by  whatever  birthday  fell  nearest  to  September) of  that 
particular  school  year.  This  meant  that  a  child  was  the 
same  age  for  a  whole  year.  Children  entering  in  June 
were  set  down  as  being  the  same  age  as  they  would  have 
been  if  they  had  entered  the  preceding  September.  This 
made  the  average  ages  of  the  children  somewhat  younger 
than  they  should  be.  It  had  the  effect  of  making  many 
children  that  were  over-age  in  the  fall,  no  longer  over-age 

1  Reported  by  Mr.  W.  F.  Puckett 


6  School  Statistics  and  Publicity 

when  promoted  at  the  mid-term.  Thus  at  the  beginning 
of  the  second  term  there  was  always  a  large  and  instan- 
taneous drop  in  the  percentage f  of  retarded  children,  al- 
though the  children  in  reality  were  relatively  about  as 
they  were  in  September.  That  is,  they  were  a  half  year 
older  and  a  half  year  farther  along  in  the  grades.  Be- 
cause of  this  peculiar  definition  of  age,  the  survey  com- 
mission could  make  no  comparisons  in  acceleration,  elim- 
ination, and  retardation,  with  other  cities.1  Similar 
difficulties  confront  a  superintendent  who  figures  retarda- 
tion before  promotions  in  his  system  one  year,  and  after 
promotions  the  next  year.  And  when  he  compares  his 
school  system  with  any  other  system  on  retardation,  he 
must  be  certain  that  he  is  taking  it  at  the  same  time  it 
was  figured  in  the  other  systems.  Another  example  is 
the  custom  reported  from  Louisville,  Kentucky,  of  count- 
ing withdrawals  as  of  the  age  they  were  on  the  previous 
September  first.  Many  of  those  withdrawing  after  the 
middle  of  the  year  are,  of  course,  in  a  higher  grade  or  half- 
grade  from  what  they  were  when  their  ages  were  taken. 
Teachers'  Salaries.  It  is  not  uncommon  to  publish 
the  salaries  of  teachers  as  being  so  much  per  month,  with- 
out any  mention  of  the  number  of  monthly  payments 
per  year.  Obviously  the  teacher  has  to  live  twelve  months 
in  the  year,  and  for  purposes  of  comparison  the  total  for 
the  year  is  the  figure  that  must  be  used,  especially  if  other 
people  are  to  be  influenced.  For  instance,  a  Kentucky 
superintendent  in  a  small  city  reported  that  when  he  got 
his  first  month's  salary  check  for  $125,  the  cashier  of  the 
bank,  who  drew  $90  a  month,  remarked  jocularly  that 
his  own  salary  should  be  raised.  He  considered  that  the 
superintendent  was  earning  more  money.  In  fact,  the 
1  Cleveland  Survey,  volume  on  "Child  Accounting,"  pp.  40-43 


Why  We  Need  Better  School  Statistics      7 

superintendent  drew  $125  a  month  for  eight  months  or  a 
total  of  $1000 ;  the  cashier  drew  $90  a  month  or  a  total 
of  $1080.  It  is  equally  obvious  that  such  a  report  on  the 
monthly  basis  is  very  much  to  the  advantage  of  the  sys- 
tem that  runs  for  a  shorter  term. 

Another  example  is  the  published  salary  list  of  the 
faculty  in  a  certain  southern  normal  school.  Some  men 
were  listed  at  $150  and  others  at  $133^  a  month.  But 
as  a  matter  of  fact,  the  $150  men  were  paid  for  eleven 
months  and  the  $133|  men  for  twelve  months  a  year, 
while  all  worked  the  same  length  of  time,  thus  making 
an  average  difference  of  only  a  little  over  $4  a  month. 

Teachers'  Schedules.  It  is  rather  common  to  attempt 
to  compare  the  amount  of  work  done  by  teachers  within 
the  same  high  school,  normal  school,  or  college,  mainly 
by  taking  simply  the  number  of  recitations  each  has  per 
week.  This  assumes  that  one  recitation  means  as  much 
work  as  any  other  and  that  teachers  do  no  work  except 
that  connected  with  recitations.  It  leaves  out  of  account 
such  factors  as :  one  recitation  with  a  large  class  means 
much  more  work  than  one  with  a  small  class ;  two  reci- 
tations in  different  classes  involve  much  more  labor  than 
two  sections  of  the  same  class ;  teachers  serving  on  cer- 
tain committees  do  a  great  deal  of  work  connected  with 
them  and  entirely  apart  from  class  duties. 

Tax  Rate.  Some  state  superintendents  and  many  city 
superintendents  have  published  comparisons  of  cities  or 
counties  with  other  cities  or  counties,  based  on  the  tax 
rate  for  school  purposes.  This  is  of  no  value  unless  one 
knows  the  rate  of  assessment  and  can  thus  arrive  at  the 
real  rate  of  taxation.  Tt  is  a  well-known  fact  that  there 
is  an  enormous  variation  in  the  rate  of  assessments  in 
various  cities ;    that  no  two   are  practically  the  same. 


8  School  Statistics  and  Publicity 

Yet  how  often  do  we  see  in  school  reports  long  lists  of 
figures  giving  the  school  tax  levies  of  many  cities  chosen 
almost  at  random.  But  these  do  not  impress  the  tax- 
payer. He  knows  whether  his  total  school  taxes  are  heavy 
or  not  and  will  pay  little  attention  to  figures  that  merely 
place  his  city  low  in  the  list  of  cities  ranked  according  to 
rate  of  school  tax.1 

But  even  the  actual  rate  of  taxation  is  still  of  little 
value  unless  one  knows  how  heavily  the  taxpayer  is  bur- 
dened with  other  taxes.  For  example,  several  years 
ago  at  Arkadelphia,  Arkansas,  the  taxpayers  protested 
against  a  school  tax  of  seven  mills  (70  cents  on  the  $100) ; 
they  said  that  if  this  were  added  to  their  state  school  levy 
of  three  mills  (30  cents  on  the  $100),  it  would  make  their 
total  school  tax  ten  mills  (100  cents  on  the  $100),  or  higher 
than  most  places  in  the  South.  This  was  totally  wrong, 
for  many  of  the  communities  with  which  they  were  com- 
paring themselves  paid  a  county  school  tax  in  addition  to 
the  local  and  state  school  taxes  considered  by  them.2 

College  Degrees.  A  similar  trouble  arises  from  the 
custom  in  many  schools  of  at  least  secondary  grade,  of 
advertising  the  percentage  of  their  faculty  that  hold  de- 
grees. But  with  such  degrees  undefined  these  figures 
mean  nothing.  In  all  sincerity  may  not  the  public  ask 
why  an  A.B.  from  Harvard  or  Vanderbilt  University  is 
rated  just  the  same  as  one  from  a  private  college  of  junior 
rank  which  has  perhaps  four  second-rate  teachers  and 
practically  no  equipment?  The  fact  is  that  the  public 
does  ask  just  such  a  question.  For  this  reason  thinking 
people  place  very  little  value  at  the  present  time  on  the 
mere  announcement  of  a  degree.  They  wish  to  know 
what    institution  conferred  it.     As  a  consequence  some 

1  Se3  pp.  51-2  -  Reported  by  Mr.  A.  E.  Phillips 


Why  We  Need  Better  School  Statistics      9 

editors,  authors,  and  catalogues,  when  giving  a  man's 
degrees,  also  indicate  the  institutions  conferring  them. 

Unequal  Things.  In  the  preceding  examples  the  lack 
of  suitable  units  has  come  about  rather  from  thoughtless- 
ness than  from  any  definite  intention.  But  a  very  com- 
mon error  in  school  statistics  is  caused  by  deliberately 
considering  as  equal,  things  which  a  very  little  serious 
thought  would  show  are  not  equal.  Professor  Thorn- 
dike  x  gives  a  good  example  of  the  seriousness  of  this 
error.  Dr.  Rice  had  ranked  a  large  list  of  words  as  of 
equal  difficulty  to  spell.  Professor  Thorndike  gave  the 
same  list  to  a  group  of  children  with  these  results :  42 
children  missed  necessary;  37,  disappoint;  and  so  on,  down 
to  1,  feather;  none,  picture.  Making  the  most  of  the 
possibility  that  the  group  of  children  found  some  of  the 
words  easier  than  others  because  of  recent  drills,  yet  the 
fact  is  unmistakably  clear  that  picture  is  not  so  difficult  for 
these  children  to  spell  as  is  necessary. 

An  especially  good  example  of  considering  as  equal 
things  which  are  not  at  all  equal  is  found  in  the 
following  excerpt  from  a  recent  circular  published  in 
Virginia : 

The  general  administration  of  school  affairs  in  Virginia  has  also 
furnished  striking  evidence  of  sound  economy  as  well  as  splendid 
efficiency.  This  is  best  shown,  perhaps,  in  the  recent  comparison 
with  the  wealthy  state  of  Minnesota,  which  has  practically  the  same 
population  as  Virginia.  In  1916  Virginia  with  eight  millions  of  rev- 
enue enrolled  11,560  more  pupils  than  Minnesota  with  twenty-six 
millions  of  revenue. 

This  ignores  the  proportion  of  children  in  the  two  state 
populations,  the  length  of  school  term,  or  anything  con- 
nected with  quality  of  education.     The  schooling  given 
1  Thorndike,  E.  L. :   Mental  and  Social  Measurements,  p.  8 


10  School  Statistics  and  Publicity 

the  average  child  in  Virginia  certainly  is  nothing  like 
the  schooling  given  the  average  child  in  Minnesota. 
The  latter  is  so  much  superior  that  it  might  con- 
ceivably cost  over  twice  as  much  and  still  be  really  more 
economical. 

A  third  instance  is  found  in  library  reports  of  various 
schools.  These  are  often  useless  for  comparative  purposes 
because  some  schools  report  all  bulletins,  agricultural 
and  census  reports,  and  duplicate  copies,  as  separate  books 
and  of  equal  value. 

A  fourth  example  is  that  furnished  by  the  vicious  at- 
tacks on  the  state  university  made  in  some  states  by  un- 
scrupulous politicians.  In  these  the  yearly  cost  of  school- 
ing for  a  child  in  a  rural  school  is  compared  with  the  several 
hundred  dollars  required  to  pay  for  a  year's  instruction 
of  a  university  student.  For  this  comparison  one  child 
in  a  rural  school  is  deliberately  considered  to  be  equal  to 
one  student  in  the  university.1 

Grading  Standards.  In  this  connection  we  must  not 
overlook  the  errors  that  arise  from  differences  in  the 
grading  standards  of  different  teachers  in  the  same  school. 
Superintendents  for  purposes  of  records,  determination  of 
class  honors,  etc.,  generally  assume  that  their  teachers 
grade  on  something  like  the  same  standard.  But,  in 
fact,  the  almost  countless  articles  and  books  on  uniform 
grading  published  in  the  last  few  years  demonstrate  that 
all  teachers  if  left  to  themselves  will  vary  widely  in  their 
standards  of  marking  pupils. 

In  some  schools  elaborate  statistics  are  kept  to  deter- 
mine the  standing  of  pupils  by  marks,  especially  in  high 
schools  for  class  or  graduating  honoi^y,  etc.  But  these 
awards  are  really  doubtful  because  no  effort  has  been  made 
1  See  also  p.  47 


Why  We  Need  Better  School  Statistics     11 

to  have  the  teachers  mark  uniformly.  It  is  only  when 
teachers  mark  on  something  like  a  uniform  basis  that  the 
valedictorian  may  be  easily  found  by  averaging  marks 
given  by  them.  Without  some  system  of  uniform  mark- 
ing, a  student  getting  an  average  of  95  may  really  be  of 
the  same  ability  as  another  getting  an  average  of  97.  In 
the  University  of  Missouri,  before  the  adoption  of  its 
present  system  of  uniform  grading,  it  was  well  known 
that  certain  departments  were  "  snap "  departments 
because  of  the  high  grades  given  in  proportion  to  the 
amount  of  work  required  of  the  student.  Professor 
Meyer  in  one  of  his  earlier  articles  on  uniform  grading  1 
shows  that  in  the  department  of  philosophy  at  this  time 
55  per  cent  of  all  grades  given  were  "A,"  while  in  the 
department  of  chemistry  only  one  per  cent  were  "A." 
Consequently,  students  of  the  university  who  wished  to 
get  into  the  Phi  Beta  Kappa  fraternity  tended  to  choose 
all  their  electives  from  the  departments  known  to  give 
many  high  marks.  It  may  be  taken  for  granted  that  few 
possessors  of  a  Phi  Beta  Kappa  key  in  this  institution 
about  this  time  had  taken  any  more  than  was  required 
in  chemistry,  but  probably  many  of  them  had  elected 
work  rather  freely  in  philosophy.  The  application  of 
the  same  principle  to  the  high  school  valedictorian  needs 
no  elaboration. 

Judging  Contests.  The  same  error  is  made  in  judging 
debating  or  essay  contests  with  from,  say  three  to  five, 
judges,  unless  the  judges  are  obliged  to  grade  on  the  same 
scale.  Suppose  we  have  ten  contestants,  and  three  judges 
who  grade  in  terms  of  their  own  judgment,  the  grades  being 
as  follows : 

1  Meyer,  Max:  "The  Grading  of  Students,"  Science,  28:  246. 
(Aug.  21,  1908) 


12 


School  Statistics  and  Publicity 


Contestant 

Marked  by 

Marked  by 

Marked  by 

Average 

Judge  A 

Judge  B 

Judge  C 

Mark 

1 

55 

82 

90 

75.6 

2 

95 

80 

94 

89.6 

3 

70 

82 

92 

81.3 

4 

85 

88 

94 

89.0 

5 

70 

82 

91 

81.0 

6 

84 

86 

96 

88.6 

7 

80 

86 

96 

87.3 

8 

75 

84 

96 

85.0 

9 

75 

80 

96 

83.6 

10 

60 

82 

92 

78.0 

Each  of  the  judges  has  rated  a  different  contestant  as  the 
highest,  and  the  proposition  is  made  that  an  average  be 
taken  of  all  the  ratings.  When  this  is  done,  it  is  found 
that  Contestant  2  has  won,  although  Judge  B  has  rated 
him  as  one  of  the  worst  of  the  performers  and  Judge  C 
has  ranked  him  as  tied  for  fifth  place.  Contestant  4 
loses  although  he  is  rated  in  first  place  by  Judge  B,  second 
place  by  Judge  A,  and  as  tied  for  second  place  in  Judge 
C's  opinion.  The  reason  he  loses,  although  his  combined 
rankings  are  higher,  is  this :  the  judges  did  not  grade  on 
the  same  scale.  Judge  A  used  a  scale  oe  much  wider 
variation  than  did  the  other  judges,  and  he  gave  Contest- 
ant 2  ten  points  more  than  any  of  the  other  contestants. 
The  whole  range  of  Judge  B's  grades  was  only  eight  points  ; 
that  of  Judge  C's  only  six  points;  but  Judge  A's  ranged 
forty  points.  Therefore,  his  rating  of  Contestant  2  more 
than  overcame  the  combined  judgments  of  the  other  two 


men. 


1  For  further  treatment  of  units  and  scales,  see  pp.  43-57. 


Why  We  Need  Better  School  Statistics     13 

3.    Comparisons  Using  Unsound  Elementary  Treatment 

Errors  in  the  Use  of  Percentages.  Of  the  unsound 
methods  of  comparison,  some  common  ones  deal  with 
percentages.  First,  there  is  the  kind  that  ignores  the 
real  meaning  of  percentage.  For  example,  a  superin- 
tendent may  claim  a  decrease  of  200  per  cent  in  retarded 
children  or  number  of  withdrawals.  Such  a  statement  is 
of  course  meaningless,  for  there  can  be  only  100  per  cent 
of  anything  on  which  to  figure  the  decrease.  What  is 
usually  meant  is  that  the  present  number  is  now  only  one 
third  of  what  it  once  was,  a  decrease  of  66f  per  cent. 

This  error  runs  at  once  into  the  one  which  frankly  loses 
sight  of  the  original  data.  Some  years  ago  a  prominent 
city  superintendent  in  Georgia  had  the  charge  brought 
against  him  that  illiteracy  of  adult  whites  had  increased 
100  per  cent  in  his  city,  while  in  the  state  as  a  whole  it 
had  decreased  from  13  per  cent  to  10  per  cent  or  a  23  per 
cent  actual  decrease  in  the  amount  of  illiteracy.  (13  -  10 
=  3.  T33  =23  %  +.)  The  facts  were  that  this  city  of  about 
forty  thousand  population  had,  at  the  beginning  of  the 
period  in  question,  a  certain  small  number  of  adult  white 
illiterates,  say  ten.  During  the  period  several  families 
of  ignorant  Greeks  moved  in,  bringing  at  least  ten  more 
adult  white  illiterates.  Based  on  the  original  number 
of  illiterates,  the  increase  was  100  per  cent;  but  based 
on  the  total  population,  the  increase  was  negligible.  Yet 
some  supposedly  well-informed  people  in  this  city,  not 
knowing  or  thinking  how  few  cases  the  misleading  state- 
ment was  based  upon,  were  inclined  to  criticize  the  super- 
intendent of  schools. 

Especially  may  the  original  data  be  forgotten  in  figuring 
percentages  on  parts  of  a  small  group.     Thus,  taking  the 


14  School  Statistics  and  Publicity 

percentages  of  the  various  marks  given  by  a  teacher  in  a 
class  of  less  than  twenty  pupils  is  of  doubtful  value.  Any 
unusual  condition  of  any  sort  affecting  one  pupil  only 
will  affect  the  number  of  pupils  receiving  the  same  mark, 
by  more  than  one  twentieth  of  the  whole.  If  there  are 
ten  pupils  in  the  class,  it  will  be  affected  10  per  cent  and 
so  on.  If  two  pupils  out  of  ten  received  A,  20  per  cent  of 
the  class  would  get  this  mark.  If  one  of  these  pupils  for 
any  reason  should  do  poorer  work,  the  A  group  would 
lose  10  per  cent  of  the  whole,  but  half,  or  50  per  cent,  of 
itself. 

In  similar  fashion  the  Missouri  School  Journal  some 
years  ago  ran  a  comparison  of  state  teachers'  associations, 
based  upon  the  percentage  of  teachers  in  the  state  attend- 
ing them.  Rhode  Island  came  first  and  Tennessee  last. 
This  frankly  ignored  to  a  considerable  extent  the  original 
data  with  its  lack  of  a  unit  for  "  teacher  in  attendance." 
The  Rhode  Island  teachers  could  attend  their  meeting 
at  Providence  on  a  nickel  street  car  fare  or  a  dime  at  the 
highest.  But  many  teachers  in  the  larger  states  could  not 
attend  their  state  association  for  financial  reasons.  Thus 
a  low  percentage  of  attendance  in  some  states  might  mean 
a  much  greater  spirit  and  devotion  than  would  a  100  per 
cent  attendance  in  Rhode  Island. 

Other  comparisons  attempt  to  relate  directly  groups  of 
data  which  should  first  have  their  component  parts  turned 
into  percentages.  For  example,  on  the  wall  in  a  certain 
state  department  of  education,  the  grades  on  certificate 
examinations  made  in  each  subject  are  plotted  in  a  graph 
ranging  from  70  to  100.  The  name  of  the  subject  is  put 
on  the  graph  and  the  name  of  the  person  grading  the 
papers.  The  attempt  is  made  to  standardize  the  grading 
in  the  different  subjects  by  comparing  these  graphs.     But 


Why  We  Need  Better  School  Statistics     15 

because  of  the  failure  to  turn  the  numbers  of  grades  for 
each  group  into  percentages  of  the  whole,  the  compari- 
sons are  either  very  difficult  or  erroneous.  The  compari- 
sons could  be  made  directly  only  if  there  were  the  same 
number  of  papers  in  each  subject,  a  thing  which  does  not 
often  happen  even  approximately. 

Carelessness  in  Securing  Data.  A  second  form  of  un- 
sound comparisons  frequently  comes  from  carelessness  in 
securing  data.  For  example,  in  comparing  school  sys- 
tems on  expenditures,  the  figures  taken  for  one  year  may 
be  wholly  misleading  for  at  least  one  or  two  cities  out  of 
any  twenty.  An  unusual  condition  in  some  one  city,  such 
as  an  epidemic  or  a  great  expansion  of  work,  or  the  erec- 
tion of  a  building  a  few  years  before,  may  make  the  ex- 
penditures for  that  year  wholly  different  from  the  usual 
expenditures  of  that  city.  Again,  unless  great  care  is 
taken,  the  total  expenditures  for  one  city  may  include 
money  spent  on  buildings  and  repairs,  and  duplicates, 
while  for  other  cities  they  cover  current  expenses  only 
without  duplicates.  Unless  the  figures  for  each  city  are 
usual,  or  "  average  "  ones,  and  unless  they  are  taken  for 
all  cities  on  the  same  basis,  no  amount  of  care  later  will 
give  sound  comparisons. 

Omission  of  Important  Factors.  A  particularly  un- 
fortunate form  of  comparison  is  that  which  presents  data 
related  to  each  other  and  then  draws  from  them  conclu- 
sions that  are  entirely  erroneous  because  certain  impor- 
tant factors  have  been  left  out  of  consideration.  In  other 
words,  it  is  the  old  fallacy  of  arguing  from  insufficient 
data.  For  instance,  take  the  case  of  the  state  superin- 
tendent in  the  South  who  some  years  ago  claimed  credit 
for  the  increase  in  school  revenues  in  his  state.  Both 
from  the  platform  and  in  press  reports,  he  claimed  that  he 


16  School  Statistics  and  Publicity 

had  done  more  towards  adding  money  to  the  state  school 
fund  than  had  any  of  his  predecessors.  He  even  issued 
a  bulletin  in  which  he  compared  the  amount  per  capita 
for  schools  during  his  regime  with  that  of  five  of  his  prede- 
cessors in  the  office.  In  his  closing  sentence  he  styled 
himself  "  the  wizard  of  finance."  As  a  matter  of  fact, 
the  state  in  question  through  industrial  development  had 
largely  increased  its  wealth  so  that  the  same  rates  for 
school  taxation  brought  larger  school  revenues.  The 
wealth  had  increased  faster  than  the  population,  and  so 
the  per  capita  spent  on  schools  rose.  However  capable 
a  state  superintendent  of  schools  may  be,  he  can  hardly 
legitimately  claim  much  credit  for  increasing  the  wealth 
of  his  state  in  a  few  years,  especially  at  a  rate  faster  than 
the  population. 

Another  example  is  the  argument  sometimes  used 
against  compulsory  education.  This  cites  that  illiteracy 
has  decreased  faster  in  certain  southern  states  that  have 
no  compulsory  education  laws  than  in  certain  northern 
states  that  have  had  such  laws  for  years.  But  it  leaves 
out  of  account  the  fact  that  the  smaller  northern  decrease 
in  illiteracy  is  due  to  recent  immigration  of  foreign  illit- 
erates, a  class  of  which  only  a  small  number  come  to  the 
South. 

4.    Attempts  at  Too  Great  Accuracy 

Cost  Figures.  Occasionally  time  and  effort  are  wasted 
by  the  superintendent's  going  to  great  extremes  to  show 
accuracy  in  his  figures.  This  is  particularly  unfortunate 
at  times,  because  what  is  considered  great  accuracy  may 
be  only  unnecessary  work  that  can  flfcver  make  things 
accurate.  For  instance,  a  superintendent  may  go  to 
great  length  to  calculate  the  cost  of  instruction  per  day 


Why  We  Need  Better  School  Statistics     17 

per  child  in  his  system,  for  comparison  with  similar  figures 
for  other  systems.  He  runs  his  figures  out  to  hundredths 
of  a  cent.  But  owing  to  the  trouble  in  units  previously 
mentioned,  his  results  are  really  worth  very  little.  One 
system  has  taken  the  school  census  for  the  number  of 
children ;  another,  the  enrollment ;  and  still  another, 
the  average  daily  attendance.  Unless  he  knows  that  the 
same  base  has  been  taken  for  each  system,  no  amount  of 
mechanical  accuracy  later  will  give  exact  results. 

In  many  public  presentations  of  school  statistics,  much 
of  the  effect  is  lost  by  attempting  to  be  too  accurate  in 
giving  the  figures  to  the  cent  or  the  fraction  thereof  in  all 
places.  The  mind  of  the  average  citizen,  or  even  of  an 
expert  for  that  matter,  cannot  take  in  too  many  details. 
The  exact  figures  add  nothing  to  the  impression  in  his 
mind,  and  indeed  detract  from  the  main  things.  As  an 
illustration  of  this  point,  take  Professor  Bobbitt's  expres- 
sions in  the  San  Antonio  Survey  that  "  English  costs  in 
the  neighborhood  of  $210,000  "  and  "  spelling  costs  in 
the  neighborhood  of  $40,000."  1  These  are  far  more  ef- 
fective than  if  he  had  run  the  sums  out  to  the  exact  num- 
ber of  dollars  and  cents  obtained  from  a  very  long  and 
tedious   computation. 

Standards.  A  superintendent  may  make  an  unwise 
recommendation  when  he  advocates  that  his  school 
should  reach  the  precise  figure  on  an  "  average "  or 
"  middle  figure  "  of  a  certain  table,  comparing  his  school 
with  other  systems.  In  view  of  the  well-known  chances 
for  inaccuracy  in  the  original  figures,  it  is  far  better  to 
take  for  granted  that  such  hair-splitting  accuracy  is  not 
desirable  and  to  use  a  rather  wide  standard.     Professor 

1  Bobbitt,  J.  F. :  Survey  of  San  Antonio  Public  Schools,  pp.  99, 
103 


18  School  Statistics  and  Publicity 

Bobbitt 1  does  this  in  his  "  zone  of  safety  "  standard 
which  includes  the  middle  half  of  the  group.  As  an  il- 
lustration of  the  workings  of  his  standard,  we  quote 
Table  I  on  the  costs  of  instruction  in  high  school  mathe- 
matics per  one  thousand  student  hours,  in  certain  cities 
of  the  country. 

Table  1.    Bobbitt  Table  Showing  Cost  of  Instruction  per 
1000  Student  Hours  (Mathematics) 

Cost  per 
Name  of  school  1000  stu- 

dent hours 

University  High $169 

Mishawaka,  Ind 112 

Elgin,  111 100 

Maple  Lake,  Minn 100 

Granite  City,  111 88 

East  Chicago,  Ind 82 

De  Kalb,  111 74~~ 

San  Antonio,  Tex 69 

Harvey,  111 69 

Waukegan,  111 63 

South  Bend,  Ind 62 

East  Aurora,  111 61 

Rockford,  111. '.     '.     7~ .     .     .     '.         5iT 

Booneville,  Mo 58 

Brazil,  Ind 56 

Leavenworth,  Kan. 56 

Greensburg,  Ind 54 

Morgan  Park,  111 53 

Noblesville,  Ind. 52 

Norfolk,  Neb7      .     .     .     ~     ~     .     .     ~     .     .     ,~     ~~~~  42 

Washington,  Mo 41 

Bonner  Springs,  Kan 38 

Russell,  Kan 34 

Junction  City,  Kan 33 

Mt.  Carroll,  111 *» .     .     .     .         30 

1  Bobbitt,  J.  F. :  "High  School  Costs,"  School  Review,  23  :  505-534. 
(Oct.,  1915; 


Why  We  Need  Better  School  Statistics     19 

From  this  table,  Professor  Bobbitt  would  not  say  that 
the  cost  of  teaching  high  school  mathematics  should  be 
exactly  $59 ;  but  that  it  should  be  somewhere  between 
$52  and  $74. l  This  measure  must  for  another  reason 
be  used  with  still  more  caution.  In  this  same  study  by 
the  same  method,  Professor  Bobbitt  found  the  "  zone  of 
safety  "  for  Latin  to  be  from  $54  to  $92,  while  that  for 
English  ranged  only  from  $43  to  $67.  That  is,  the  zones 
differ  for  different  subjects,  and  one  cannot  at  first  glance 
be  sure  that  one  ought  to  strive  for  a  difference  in  costs 
in  the  subjects.  Again,  this  measure  is  not  good  for 
publicity  purposes  unless  the  superintendent's  system  is 
below  the  "  safety  zone."  In  such  cases  it  is  very  effec- 
tive. But  if  his  system  is  above  this  zone,  citizens  are  apt 
to  rest  contented  or  even  to  contemplate  reducing  school 
taxes. 

5.    Neglect  of  Technical  Statistics 

Averages.  Even  when  the  data  are  accurate,  errors 
may  creep  in  because  of  neglect  of  the  elements  of  tech- 
nical statistics.  A  frequent  example  of  this  arises  from  a 
very  loose  use  of  the  average  to  typify  a  group,  especially 
in  regard  to  teachers'  salaries.  In  a  recent  investigation 2 
of  salaries  paid  by  fifteen  of  the  best-known  colleges  for 
women,  it  was  found  that  the  average  lowest  salary  paid 
was  $700  and  the  average  highest  was  $1500.  The 
average  would  give  a  very  erroneous  impression,  as  far 
more  of  the  instructors  received  the  lower  salaries  than 
the  higher.  The  maximum  amount  reported  was  $3000 ; 
the  lowest,  $100  and  home.  It  is  obvious  that  the  high 
salary  would  exert  a  greater  influence  on  the  average  than 
the  lower  one,  since  it  is  twice  as  far  away. 

1  To  be  still  more  accurate,  between  $47  and  $78.     See  p.  130. 

2  Journal  of  Pedagogy,  19:  185 


20  School  Statistics  and  Publicity 

This  point  may  be  further  illustrated  from  a  report  of 
a  state  superintendent.1  In  one  table  he  gives  the  highest 
and  the  lowest  salary  paid  per  month  to  male  and  to 
female  teachers,  for  both  races,  separately.  The  table 
following,  in  the  same  way,  gives  the  average  salary  of 
the  various  groups.  For  example,  County  A  reports 
the  highest  salary  for  white  male  teachers,  $175  per  month ; 
the  lowest,  $40;  the  average,  $73.07.  The  fact  un- 
doubtedly is  that  in  County  A  more  white  male  teachers 
get  less  than  $73.07  than  get  more.  This  table  is  made 
worse  by  averaging  the  salaries  paid  both  white  teachers 
and  colored.  This  average  means  little  for  two  reasons : 
First,  there  are  fewer  colored  teachers  by  far  in  Florida 
than  white ;  second,  the  colored  teachers  do  not  get  any- 
thing like  so  much  salary  as  the  white.  The  result  is 
that  the  average  is  higher  than  all  or  most  of  the  salaries 
paid  colored  teachers,  and  lower  than  all  or  most  of  the 
salaries  paid  white  teachers,  and  so  is  significant  for 
neither  group. 

In  the  summaries  of  another  report  of  the  same  super- 
intendent of  the  same  state  for  1913  14,  this  error  appears.2 
The  average  salary  per  month  paid  teachers  in  the  ten 
highest  and  the  ten  lowest  states  in  the  United  States  is 
given.  Wisconsin  stands  at  the  top  of  the  lowest  ten. 
Florida  does  not  appear,  so  we  suppose  that  she  is  some- 
where between  the  highest  ten  states  and  the  lowest  ten ; 
hence  the  apparent  conclusion  that  Florida  pays  a  better 
monthly  salary  than  does  Wisconsin.  But  from  another 
table  we  learn  that  the  average  school  term  in  Wisconsin 
is  175.7  days ;  for  Florida,  122.2  days.  In  other  words, 
Wisconsin  pays  on  the  average  for  53. o  more  days  than 

1  Report  of  State  Superintendent  of  Florida,  1912:  471   472 

2  Report  of  State  Superintendent  of  Florida,  1913-14  :  37 


Why  We  Need  Better  School  Statistics     21 

does  Florida.  Now  if  these  averages  had  been  expressed 
in  years  instead  of  months,  the  chances  are  very  likely 
that  Wisconsin  would  be  higher  up  in  the  list  of  states 
than  Florida. 

A  very  inaccurate  but  common  way  of  taking  the  aver- 
age is  illustrated  by  the  following  procedure  which  came 
under  the  writer's  notice :  In  a  study  for  school  purposes 
of  the  negroes  of  Texas,  the  percentage  of  the  negro  pop- 
ulation to  the  total  population  of  the  state  was  desired. 
Three  counties  with  a  high  percentage,  three  with  a  mod- 
erate percentage,  and  three  with  a  low  percentage  were 
taken  for  the  whole  state.  Then  the  average  of  these  nine 
counties  was  determined.  This  is  a  very  rough  and  in- 
accurate method.  If  the  number  of  counties  in  the  state 
were  divided  into  groups  having  the  same  ratio  to  each 
other  as  the  counties  taken  in  the  study,  and  if  these  latter 
were  then  chosen  at  random  or  equally  spaced  in  their 
groups,  the  results  would  be  fairly  accurate.  But  such  a 
state  of  affairs  could  hardly  be  hoped  for  in  popular 
sampling  of  this  kind. 

A  similar  error  is  liable  to  occur  in  the  common  practice 
of  judging  the  work  of  a  class  by  examining  one  of  the 
best  notebooks,  one  of  the  medium  group,  and  one  of 
the  worst.  Unless  care  is  taken,  the  extremes  may  re- 
ceive an  undue  amount  of  emphasis.  The  medium  group 
is  always  so  much  larger  than  the  extreme  groups  that  it 
ought  to  have  at  least  two  samples  to  each  one  for  the 
other  groups. 

Variations.  Often  bare  averages  are  given,  when  varia- 
tions from  the  average  are  the  significant  thing.  Suppose 
that  two  boys  in  school  have  the  same  average  for  their 
grades  in  different  subjects,  say  85.  But  one  boy  made 
these  grades,  84,  86,  86,  88,  81,  and  the  other  made  97,  98, 


22  School  Statistics  and  Publicity 

yO,  80,  60.  The  deviations  from  the  average  in  these  two 
instances  show  that  one  boy  is  an  "  average  "  boy  in  all 
his  studies  while  the  other  is  very  fine  in  some,  possibly 
those  he  likes,  but  poor  in  the  others. 

In  the  illustration  of  the  salaries  paid  teachers  in  col- 
leges for  women,  previously  given,  the  significant  thing 
is  not  the  average  but  the  deviation  from  that  average. 

Professor  Bobbitt  in  his  San  Antonio  survey  has  one 
rather  incomplete  treatment  in  this  matter  of  deviation, 
occurring  in  what  is  otherwise  a  most  admirable  use  of 
variations.1  He  desires  to  make  the  point  that  the 
problem  of  heating  the  schoolrooms  in  the  city  is  of  minor 
importance  and  gives  a  table  of  mean  hourly  temperature 
of  the  city  for  all  school  hours,  month  by  month.  This 
of  course  allows  for  deviations.  These  figures  are  for  the 
winter  months  not  a  tremendous  distance  below  the  68 
degree  standard  for  comfort.  This  would  indicate  that 
the  city  was  always  mild  in  winter.  However,  the  sig- 
nificant thing  is  not  this  mean  temperature  of  San  An- 
tonio, but  the  deviations  from  it.  To  get  a  mean  temper- 
ature of  say  55,  there  must  be  days  below  it,  possibly  a 
considerable  number.  If  there  are  only  a  few  days  each 
month,  it  is  clear  that  the  school  buildings  must  be 
equipped  to  keep  children  warm  on  those  days  or  else 
school  must  be  dismissed  until  it  gets  warmer. 

Sampling.  An  extremely  common  error  in  school 
statistics  arises  because  of  ignorance  of  "  sampling." 
This  corresponds  to  the  error  in  logic  arising  from  con- 
clusions drawn  from  one  case,  from  too  few  cases,  or  from 
cases  not  properly  selected.  In  many  school  statistics 
the  samples  are  too  few,  or  they  are  noTlmpartially  taken. 
Especially  is  this  true  in  most  questionnaire  or  straw 
1  Sun  Antonio  Survey,  pp.  226  ff. 


Why  We  Need  Better  School  Statistics     23 

ballot  investigations.  A  questionnaire  is  generally  an- 
swered chiefly  by  those  especially  interested  in  the  subject. 
Any  conclusions  from  such  data  should  not  be  interpreted 
as  applying  to  any  save  that  class.  The  methods  of  taking 
the  average  described  on  page  21  are  of  course  illustra- 
tions of  bad  sampling. 

Incorrect  sampling  appears  often  in  school  advertising. 
Thus,  a  leading  southern  university  gives  prominence  in 
its  advertising  to  the  number  of  its  graduates  who  have 
been  members  of  Congress,  governors  of  states,  United 
States  senators,  or  even  President.  In  many  ways  of 
course  such  advertising  is  thoroughly  justifiable,  but  not 
when  it  seeks  to  convey  the  impression  that  the  typical 
student  at  the  university  will  later  reach  such  prominence. 
Again,  a  certain  southern  university  appears  to  admit  to 
graduate  standing  all  graduates  of  a  certain  normal 
school  which  does  about  two  years  of  college  work.  This 
graduate  standing  is  granted  because  at  one  time  three  or 
four  students  from  the  normal  school  in  question  were 
taken  on  trial  at  this  university  and  did  good  graduate 
work.  That  is,  all  students  from  the  normal  school  are 
assumed  to  be  as  capable  as  the  original  three  or  four. 
Practically  every  school  system,  especially  a  high  school 
striving  for  recognition,  cites  the  performance  of  its  best 
students  in  creating  a  general  impression  of  its  work. 
But  such  sampling  needs  to  be  taken  with  considerable 
salt.  We  might  with  equal  justice  expect  all  Chinamen 
to  be  so  many  Confuciuses,  or  all  Americans  to  be  so  many 
Woodrow  Wilsons. 

The  process  of  sampling  is  treated  at  length  later  on.1 
But  briefly,  where  sampling  is  resorted  to,  it  must  be  done 
purely  at  random.     And  enough  cases  must  be  taken  to 

1  Pp.  62-71 


24  School  Statistics  and  Publicity 

insure  an  approximately  accurate  impression  of  the  entire 
distribution  that  is  being  sampled. 

Number  on  a  Scale.  A  final  technical  weakness,  one 
requiring  great  care  to  avoid,  comes  from  a  wrong  inter- 
pretation of  what  a  number  means  on  a  scale.  For  ex- 
ample, a  score  of  6.25  on  any  test  should  mean  one-fourth 
of  a  step  above  the  "  6  "  step's  starting  point.  If  this 
step  runs  from  6  to  7,  obviously  the  score  of  6.25  is  just 
what  we  need.  If  "  6  "  is  regarded  as  running  from  5.5 
to  6.5,  then  6.25  may  really  mean  5.75.  A  superin- 
tendent may  measure  his  work  on  a  scale  by  one  method 
and  compare  it  with  work  measured  by  another  method 
on  the  same  scale,  and  feel  much  elated  over  beating  the 
other  man  by  half  a  step,  when  as  a  matter  of  fact  the 
half  step  advantage  is  due  solely  to  differences  in  counting 
on  the  scale.1 

6.   Presentations  of  School  Statistics  to  the  Public 

When  it  comes  to  presenting  school  statistics  to  the 
public,  the  errors  are  even  greater.  There  is  often  little 
or  no  tabulation,  or  there  are  such  complications  that 
they  can  be  understood  only  by  experts.  Many  of  the 
presentations  make  no  use  of  graphs,  and  others  contain 
such  intricate  graphs  or  charts  that  laymen  cannot  easily 
understand  them.  Some  of  the  charts  have  the  zero  line 
cut  off  so  that  the  effect  on  the  reader  is  apt  to  be  totally 
misleading.2  In  other  graphs  the  scales  used  may  give  a 
false  impression.3 

II.   WASTEFULNESS   OF   THESE   ERRORS 

The  mere  errors  of  school  statistics  are  not  their  worst 
feature  ;  it  is  the  amount  of  time  and  energy  put  into  such 

1  See  p.  49  =  See  p.  2S3  :i  See  p.  284 


Why  We  Need  Better  School  Statistics     25 

useless  things.  Some  school  executives  apparently  de- 
vote much  time  and  effort  to  the  collection  and  publica- 
tion of  educational  statistics  of  this  useless  sort,  pre- 
sumably with  the  idea  of  doing  some  good.  Professor 
Hanus  J  in  a  recent  study  of  superintendents'  reports 
found  this  to  be  true  of  one  city :  The  report  of  the  super- 
intendent contained  35  tables,  23  of  which  contained 
statistics  for  the  year  1913-14;  the  other  12  presented 
comparative  statistical  summaries  covering  the  period 
from  1906  to  1914 ;  but  only  two  of  the  latter  group  of 
tables  had  any  relation  to  the  first-named  group.  Forty- 
two  per  cent  of  all  the  reports  studied  by  Hanus  con- 
tained "  few  or  no  comparative  statistics,  and  very 
little  or  no  satisfactory  interpretation  of  statistics." 
The  significance  of  such  statistics  is  practically  nil,  he 
concluded. 

Snedden  and  Allen,  in  a  book  published  some  years  ago,2 
indicate  that  in  their  investigation  of  the  school  reports 
of  the  best  city  systems  of  the  country,  much  useless 
statistical  material  was  found.  This  material  they  roughly 
classified  as  of  two  kinds :  First,  material  that  would  be 
useful  if  properly  related  and  explained ;  second,  material 
useless  in  itself,  either  because  it  has  no  significance  what- 
ever, or  because  it  is  in  detailed  form,  when  only  the  sum- 
mary is  the  significant  thing. 

The  protests  against  such  ineffectiveness  in  school 
statistics  arise  also  from  other  sources.  Superintendent 
Maxwell  of  New  York  frankly  says :  "  There  are  some 
ways  in  which  the  efficiency  of  a  school  may  be  deter- 
mined with  an  approach  to  accuracy  and  without  the 

1  Hanus,  Paul :  "Town  and  City  Reports,"  etc.,  School  and  Society, 
3 :  145-155,  186-198.     (Jan.  29,  Feb.  5,  1916) 

2  School  Reports  and  School  Efficiency,  pp.  1-10 


26  School  Statistics  and  Publicity 

assistance  and  without  the  retardation  of  time*wasting, 
energy-destroying  statistical  research. 

"  There  may  be  ways  in  which  the  so-called  scientific 
surveys  or  investigations,  when  stripped  of  past  and 
present  absurdities,  will  help  in  determining  efficiency."  1 

Again,  the  writer  of  a  very  thought-provoking  satire 
on  present-day  educational  theory  offers  the  following 
"  statistics  on  the  different  classes  of  teachers  with  re- 
spect to  '  pedaguese  '  or  the  scientific  terminology  em- 
ployed by  educational  theorists  " 2 : 

Use  and  think  they  understand  it        12% 

Have  used  and  thought  they  understood  it,  but  don't  now     .  2% 

Think  they  understand  it,  but  don't  use  it 6% 

Use  it  but  don't  understand  it 9% 

Don't  use  it,  don't  understand  it,  but  esteem  with  awe  those 

who  do 51% 

Think  it  is  rot 20% 

100% 
III.    INDIFFERENCE    OF   THE   PUBLIC 

Moreover,  the  public,  particularly  the  influential 
portion  of  the  public,  have  a  distrust  of  these  school 
statistics  and  are  little  affected  by  them.  It  may  be  that 
the  average  citizen  has  a  wholesome  respect  akin  to  awe 
for  any  statistical  presentation,  as  claimed  by  some 
writer  in  the  Unpopular  Review*  In  this  article  appear 
some  very  illuminating  examples  of  misleading  statistics 
in  general  fields.  Errors  similar  to  these  can  be  found 
in  many  school  reports.  But  the  writer's  experience 
and  results  from  conferring  with  many  practical  school 

men    point    in    the    other    direction.     Many    influential 

*■» 

1  Maxwell,  W.  H. :  "How  to  Determine  the  Efficiency  of  a  School 
System,"  American.  School  Board  Journal,  May,  1915 

2  Henrick,  Welland  :   A  Joysome  HiMorjj  of  Education,  p.  54 
;  May-June,   191  5  :  352-366 


Why  We  Need  Better  School  Statistics     27 

people  who  are  not  especially  trained  in  statistics,  are 
frankly  skeptical  of  any  statistical  report  on  schools.  At 
the  same  time  they  may  be  vitally  interested  in  the  schools 
and  have  confidence  in  the  superintendent. 

School  statistics  are  usually  included  in  large  reports 
which  do  not  reach  the  public  to  any  appreciable  extent. 
Professor  Hanus  sent  250  letters  to  the  members  of  the 
Harvard  Club,  and  250  to  the  members  of  the  Chamber 
of  Commerce  in  Boston.  He  asked  these  people  if  they 
had  seen  a  report  of  the  school  board,  including  that  of 
the  city  superintendent,  within  the  last  two  years.  He 
received  128  replies  from  members  of  the  club  and  83 
from  members  of  the  Chamber  of  Commerce.  About 
70  per  cent  of  these  answered  "  No."  Snedden  and  Allen 
assert  that  the  reports  of  city  systems  are  not  read  and 
give  as  the  reason  that  they  are  not  so  arranged  as  to  be 
intelligible  to  the  ordinary  citizen.  The  common  practice 
of  issuing  summaries  or  abstracts  of  school  reports  or 
surveys,  indicates  the  same  thing. 

The  indifference  of  the  public  is  further  shown  by  the 
fact  that  editions  of  school  reports  are  usually  very  small. 
Only  a  few  years  ago  the  proposition  was  made  to  place 
copies  of  the  New  York  City  report  in  the  hands  of  each 
of  the  15,000  teachers,  so  that  more  intelligent  and  ear- 
nest work  would  result.  It  was  objected  to  on  the  ground 
that  it  would  cost  too  much.  But  the  cost  of  the  reprints 
would  have  been  insignificant  compared  with  the  $36,000,- 
000  spent  on  the  public  schools  that  year.  Another 
illustration  comes  from  Mobile.  This  city  has  a  debt  of 
$4,000,000,  and  an  annual  expenditure  of  about  $464,000 
with  $110,000  spent  on  schools.  But  the  board  hesi- 
tated about  paying  the  $500  necessary  to  print  the  recent 
school  survey. 


28  School  Statistics  and  Publicity 

IV.    PROGRESS   ELSEWHERE   IN   USING    STATISTICS 

While  these  and  similar  errors  continue  in  statistics 
of  many  school  systems  with  resulting  indifference  of  the 
public,  remarkable  improvements  and  extensions  in  the 
use  of  statistical  method  have  been  made  in  other  fields 
and  in  some  schools.  Whether  we  look  in  economics, 
sociology,  census  reports,  insurance,  cost  accounting, 
biological  sciences,  or  the  best  school  reports,  wherever 
scientific  method  is  used,  that  is,  practically,  wherever  the 
experimental  method  is  used,  there  has  been  a  refinement 
of  statistical  method  and  an  application  of  it  to  the  prob- 
lems in  question. 

Let  us  list  at  random  some  of  these  problems  as  they 
come  to  mind : 

Life  insurance  calculations,  by  which  the  average  number  of  years 
a  man  will  live  after  any  given  age,  is  known. 

Weather  conditions  (rain,  temperature,  etc.)  over  any  given  area 
for  a  long  period  of  time,  by  which  it  may  be  known  whether  it  will 
be  profitable  to  grow  certain  crops  in  that  region. 

The  adaptation  of  the  average  to  measuring  and  sampling  processes, 
especially  of  grain. 

The  movement  from  country  to  city  and  its  results. 

The  tenancy  problem,  its  causes,  types,  and  effects  on  general 
social  status  of  the  region  under  consideration. 

Relative  attainments  of  city  and  country  born  children. 

Relative  strength  of  various  European  belligerents. 

The  relation  of  men  and  women  in  various  mental  processes. 

The  effect  of  factory  labor  on  children. 

The  fight  against  preventable  diseases. 

Comparison  of  the  intellect  of  the  white  man  with  that  of  the  negro. 

The  influence  of  woman  suffrage  in  national  elections. 

Within  the  field  of  educational  investigation  itself,  it 
is  significant  to  scan  the  list  of  publications  at  a  graduate 
school  of  education  like  Teachers  College  (Columbia 
University)  and  see  that  from  only   a   small    number   of 


Why  We  Need  Better  School  Statistics     29 

dissertations  embodying  a  few  statistics  simply  handled, 
there  has  been  a  steady  drift  toward  investigations  in- 
volving the  comprehensive  use  of  statistical  method  of 
greater  and  greater  accuracy.  The  School  of  Education 
at  the  University  of  Chicago  requires  a  course  in  statis- 
tical method  very  early  in  the  program  of  every  graduate 
student. 

Statistical  method  has  been  used  to  get  at  the  truth  of 
many  educational  problems,  ones  that  could  not  other- 
wise be  solved,  for  example : 

Do  women  teachers  drive  boys  out  of  high  schools? 

Is  the  South  taxing  itself  as  high  for  schools  as  the  North? 

Is  the  record  on  college  entrance  examinations  as  good  an  index 
of  the  student's  later  achievement  in  college  as  is  his  high  school 
record  or  the  college  record  of  his  brother? 

What  is  the  relation  between  reasoning  and  fundamental  opera- 
tions in  arithmetic? 

What  are  the  most  economical  ways  of  memorizing? 

Do  teachers  get  more  salary  for  each  year  of  training  beyond  the 
high  school? 

What  are  the  causes  of  elimination  of  children  from  school  ? 

A  similar  progress  has  been  made  in  these  other  fields 
in  presenting  statistical  information  to  the  public.  Po- 
litical parties,  labor  leaders,  Y.  M.  C.  A.  leaders,  phil- 
anthropic workers,  various  foundations,  bureaus  of  mu- 
nicipal research,  evangelists,  advertising  agencies,  and 
corporations  are  showing  in  their  reports  great  progress  in 
pictorial  and  graphic  ways  of  presenting  statistical  mate- 
rial so  as  to  influence  the  layman. 

V.    UNSATISFACTORY    STATE    OF   AFFAIRS   IN    SCHOOL 
STATISTICS 

In  view  of  the  progress  in  the  use  of  statistics  in  other 
fields  and  in  special  educational  investigations,  our  present 
unsatisfactorv  state  of  general  school  statistics  cannot  be 


30  School  Statistics  and  Publicity 

allowed  to  continue.  Statistical  methods  apply  wherever 
things  are  to  be  counted  or  measured.  Nearly  all  the 
problems  of  the  school  executive  involve  numerical  data 
and  cannot  be  adequately  handled  without  statistical 
method.  For  example,  what  problem  would  a  super- 
intendent have  that  did  not  relate  to  one  of  the  following 
general  fields? 

1.  Children  to  be  educated  or  changed. 

2.  The  aims  of  education,  or  the  nature  and  amount  of  change  to 
be  produced  in  these  children. 

3.  The  agents  of  this  education  —  teachers  and  others. 

4.  The  means  of  this  education  —  buildings,  books,  laboratories, 
etc. 

5.  The  methods  by  which  these  agents  use  these  means. 

6.  The  changes  resulting  from  various  combinations  of  these  with 
the  first,  which  is  the  big  thing  in  education.1 

Every  one  of  these  fields  affords  numerical  data  for  the 
solution  of  problems,  and  they  cannot  be  solved  without 
handling  such  data.  In  such  fields  "  the  number  of  use- 
ful studies  to  be  made  is,  for  all  practical  purposes,  in- 
finite." 2 

The  school  executive  has  as  great  need  to  appeal  to  the 
layman  with  statistical  matter  on  schools,  as  do  any  of 
the  publicists  in  other  fields.  In  particular,  he  has  prac- 
tically to  make  the  same  appeal  for  funds  as  do  workers 
in  these  other  fields,  especially  the  religious  and  philan- 
thropic ones..  Consequently,  the  school  superintendent 
needs  for  school  purposes  any  valuable  methods  or  dis- 
coveries on  statistics  worked  out  in  other  fields. 

1  Thorndike,  E.  L. :  "Quantitative  Investigations  in  Education," 
School  Review  Monographs-  for  College  Teachers  of  Education,  No.  I, 
PP.  :n  32 

Ibidem,  p.  34 


Why  We  Need  Better  School  Statistics     31 

In  the  preceding,  the  aim  has  been  to  set  forth  clearly 
typical  errors  in  school  statistics  and  to  indicate  briefly 
the  causes  for  such  ineffectiveness.  The  rest  of  this  book 
shows  in  detail  how  to  work  up  school  statistics  properly 
and  how  to  present  them  effectively  to  the  public. 

Chapter  Two  deals  with  the  collection  of  data.  Chap- 
ter Three  discusses  the  need  for  technical  methods  in  han- 
dling school  statistics.  Chapters  Four,  Five,  Six,  and 
Seven  show  how  to  apply  these  technical  methods  to  school 
facts  that  are  to  be  presented  to  the  public,  —  Chapter 
Four  dealing  with  scales,  distribution  tables,  and  surfaces 
of  frequency ;  Chapter  Five  with  the  three  measures  of 
type ;  Chapter  Six  with  the  measures  of  deviation  or  dis- 
persion ;  Chapter  Seven  with  measures  of  relationship. 
Chapter  Eight  is  a  supplement  on  statistical  treatment, 
dealing  with  additional  problems  which  the  superintendent 
will  encounter  in  working  up  his  school  statistics  to  present 
to  the  public.  Chapter  Nine  treats  of  the  ineffectiveness 
with  the  public  of  the  statistics  in  many  school  reports. 
Chapter  Ten  shows  how  to  tabulate  school  statistics  for 
the  layman.  Chapter  Eleven  deals  with  graphic  presenta- 
tions of  school  statistics  for  the  public.  Chapter  Twelve 
gives  effective  ways  of  translating  school  statistics  into 
words,  for  popular  consumption. 

EXERCISES 

1.  When  may  a  child  be  counted  as  a  six-year-old? 

2.  Just  what  is  the  justice  of  a  state  university's  rating  a  high 
school  on  the  performance  of  its  former  graduates  in  the  first  two 
years  at  the  university? 

3.  Precisely  what  does  a  boast  in  the  state  teachers'  journal  that 
a  certain  girl  graduate  in  a  given  high  school  made  an  average  of  98.5 
for  the  four  years,  mean  to  other  high  school  teachers? 

4.  How   valuable   is   the   practice  of   some   teachers   of   keeping 


32  School  Statistics  and  Publicity 

records  of  the  work  of  pupils  for  only  the  last  few  days  before  the  end 
of  the  term  ? 

5.  How  significant  is  the  claim  of  a  superintendent  that  his  high 
school  is  better  than  that  of  a  neighboring  superintendent  because 
he  has  in  it  6  teachers  as  against  the  other's  4,  and  enrolls  60  non- 
residents as  against  the  other's  40? 

6.  Of  what  worth  is  the  statement  of  a  superintendent  that  he 
has  increased  the  size  of  the  graduating  class  in  his  high  school  75 
per  cent  in  two  years? 

7.  How  could  you  compare  in  the  same  high  school  the  cost  of 
teaching  one  period  a  day  for  the  principal  who  teaches  4  periods  a 
day,  with  that  for  the  English  teacher  who  has  6  periods  a  day ;  with 
that  for  a  science  teacher  who  has  4  classes  with  double  periods  each, 
a  day? 

8.  What  examples  of  the  errors  given  on  pages  2-24  have  you  come 
across  in  your  previous  experience  with  school  statistics,  and  in  just 
what  did  the  trouble  consist? 

9.  Take  some  school  report  or  some  article  in  your  state  teachers' 
journal  that  contains  considerable  statistical  material.  Locate  the 
errors,  if  any,  in  it  and  show  in  just  what  they  consist. 


CHAPTER   II 
COLLECTION    OF   DATA 

I.  WHEN  TO  USE  STATISTICAL  METHOD 
1.    When  Statistical  Method  Is  Profitable 

Which  school  problems  may  be  subjected  with  profit 
to  statistical  treatment?  The  superintendent  needs  to 
know  this,  for  we  have  seen  how  easy  it  is  for  him  to 
waste  his  time  and  make  errors  in  school  statistics. 
These  troubles  frequently  come  from  two  ever-present 
dangers :  starting  on  the  details  of  the  task  before  it 
has  been  clearly  thought  out ;  and  attempting  to  use 
statistical  method  where  it  is  not  applicable  or  will  not 
yield  results  worth  the  effort.  In  other  words,  two 
questions  at  once  arise  : 

1.  Which  problems  of  the  superintendent  may  profitably  be  sub- 
jected to  statistical  treatment? 

2.  How  may  he  know  whether  a  given  problem  can  profitably  be 
so  treated? 

In  reply  to  the  first  of  these  questions,  it  may  be  said 
that  practically  all  of  the  superintendent's  problems  may 
be  statistically  treated,  because  on  all  of  them  he  can 
work  with  data  that  may  be  counted  or  measured  more 
or  less  accurately.  The  quotation  from  Professor  Thorn- 
dike,  page  30,  shows  how  wide  a  range  of  problems  can 
be  thus  treated.  But  to  be  still  more  specific,  the  follow- 
ing list  of  problems  ever  present  for  the  superintendent 
must  be  so  treated  :  l 

1  Made  up  chiefly  from  Snedden  and  Allen :  School  Reports  and 
School  Efficiency,  pp.  118-127 

33 


34  School  Statistics  and  Publicity 

Buildings 

Is  the  number  of  sittings  adequate? 

What  is  the  value  per  child  that  can  be  comfortably  seated  ? 

How  much  has  the  city  spent  for  buildings  per  $1000  of  taxable 
property  as  compared  with  other  cities  for  a  period  of  years? 

How  much  waste  space  is  there  in  the  buildings? 

What  is  the  proportionate  cost  of  upkeep  and  repairs,  per  capita 
average  daily  attendance? 

What  has  been  the  cost  of  equipping  special  buildings  or  rooms,  as 
laboratories,  manual  training,  etc.,  per  capita  average  daily  at- 
tendance, as  compared  with  other  years  and  other  systems? 
Receipts  and  expenditures 

What  has  been  the  per  capita  expenditure,  expressed  in  terms  of 
average  daily  attendance  of  the  system  as  a  whole  for  several  years? 

What  has  been  the  per  capita  cost,  per  school,  of  such  items  as 
salaries,  general  administration,  fuel,  building  and  repairs, 
special  classes,  etc.?     Also  what  have  been  the  relative  costs? 

What  do  the  different  classes  of  schools,  —  high,  junior  high,  ele- 
mentary, special,  —  cost  per  capita? 

Are  receipts  keeping  pace  with  the  increase  in  number  of  children  ? 
Census 

What  are  the  numbers  of  children  for  each  year  within  the  school 
age  limits  in  the  city  ? 

How  many  of  these  are  within  the  compulsory  school  age  ? 
Attendance 

What  is  the  number  of  children  of  compulsory  school  age  attend- 
ing public  schools? 

How  many  children  of  compulsory  school  age  are  attending  private 
schools,  special  schools,  or  are  otherwise  satisfactorily  accounted 
for? 

How  many  are  in  voluntary  attendance,  per  school,  and  has  the 
number  increased  during  the  past  several  years? 

What  is  the  number  of  persistent  attendances,  i.e.,  children  who 
attend  160  days  out  of  a  possible  200,  etc.? 

What  is  the  character  of  the  absences?  ^ 

Elimination 

How  many  children  drop  out  each  year,  by  grades  and  schools? 

Why  do  these  children  drop  out  —  transferred  to  other  schools, 
non-promotion,  irregular  attendance,  over-age,  etc.? 

Is  the  percentage  of  elimination  increasing? 


Collection  of  Data  35 

Retardation 

What  are  the  percentages  of  promotion  and  non-promotion  by 
grades  and  schools? 

Would  retardation  be  lessened  by  flexible  promotion  schemes? 
By  promotion  by  subject  in  the  upper  grades? 

Are  retardation  and  elimination  increased  or  lessened  in  schools 
where  industrial  work  is  given? 

What  is  the  relation  between  retardation  and  over-age? 
Special  classes 

What  is  the  per  capita  cost,  average  daily  attendance,  of  special 
classes,  evening  schools,  etc.? 

How  does  this  cost  compare  with  that  in  other  cities? 
Medical  inspection 

How  many  children  have  been  treated  ?  How  many  defects  have  been 
remedied  as  a  result  of  this  treatment  ?    How  many  homes  visited  ? 

What  is  the  per  capita  cost  of  this  work  ? 

Is  it  adequate? 
Truancy 

How  many  cases  were  reported? 

What  disposition  was  made  of  these  cases? 

What  were  the  causes  of  these  cases  of  truancy? 

What  has  been  the  per  capita  cost  of  the  truancy  department  ? 

How  does  it  compare  with. other  cities  in  amount  of  work  done  and 
in  expense? 

How  much  time  on  the  average  elapses  before  a  reported  truant  is 
returned  to  school  ? 
Supervision 

Which  of  several  methods  of  teaching  a  subject  produces  the  best 
results  with  the  given  teachers  and  conditions  of  the  system? 

Which  teachers  or  schools  are  doing  the  best  work,  in  what  subjects, 
and  just  how  much  better? 

How  many  teachers  can  a  supervisor  most  profitably  work  with  in 
the  given  system? 

How  successful  is  the  scheme  of  tne  system  for  rating  teachers? 

How  effective  for  producing  better  teaching  are  the  methods  em- 
ployed for  improving  teachers  in  service? 

Are  the  teachers  overburdened  with  routine  and  clerical  work? 

Are  the  classes  too  large  for  good  work  ? 

Which  parts  of  the  curriculum  ought  to  be  eliminated  or  given  re- 
duced time  allotments? 


36  School  Statistics  and  Publicity 

2.    When  Statistical  Method  Is  Unprofitable 

Sometimes  statistical  method  cannot  be  used,  because 
it  is  impracticable  to  get  suitable  data.  Lack  of  records, 
of  suitable  help,  of  money  to  pay  for  such  assistance,  or  of 
time  on  the  part  of  the  superintendent,  may  bring  this  about. 

There  are  cases  also  in  which  the  effect  aimed  at  with 
a  statistical  presentation  could  be  secured  more  quickly 
and  easily  with  some  argument  from  analogy,  some  emo- 
tional appeal,  etc.  For  example,  one  of  the  Cleveland 
reports  some  years  ago  wished  to  emphasize  the  improve- 
ment in  school  children  that  could  be  produced  by  op- 
erations for  adenoids  and  similar  troubles.  Instead  of 
using  statistics,  it  simply  showed  the  picture  of  a  boy 
afflicted  with  adenoids  before  the  operation,  and  another 
picture  of  him  some  months  afterward.  The  difference 
in  expression  in  the  two  pictures,  especially  as  regards 
intelligence,  probably  was  more  effective  than  any 
possible  statistical  presentation. 

Then  there  are  cases  of  individuals  which,  as  Professor 
King  so  aptly  puts  it,  "  statistics  cannot  and  never  will 
be  able  to  take  into  account.  When  these  are  important, 
other  means  must  be  used  for  their  study."  l  The  second 
of  these  statements  is  really  true  only  on  the  assumption 
that  the  superintendent  has  the  right  conclusion  to  present. 
Often  this  statistical  treatment  would  show  him  that  his 
presentation  was  untrue  or  at  least  needed  material 
modification. 

Finally,  there  are  some  school  problems  that  cannot  be 
treated  statistically  with  profit,  for  examj^e : 

Cost  of  subjects  sure  to  enroll  few  students  compared  with  the  cost 
of  those  sure  to  enroll  many. 

Getting  a  city  already  spending  money  freely  to  do  better. 
1  King,  W.  I. :    Elements  of  Statistical  Method,  p.  35 


Collection  of  Data  37 

Selection  of  texts ;   cheap  books  usually  mean  inferior  quality. 

Percentages  of  elimination  and  retardation  where  it  is  impossible 
to  estimate  fairly  accurately  the  number  of  children  entering  in  a 
given  year. 

Worth  of  teaching  methods  outside  of  those  that  may  be  objec- 
tively measured.  Superintendent  Maxwell  in  the  article  previously 
cited  gives  a  few  such  cases,  for  example,  the  items  of  character- 
building,  development  of  reasoning  ability  (not  yet  measured  to  any 
appreciable  extent),  the  motive  to  good,  hard  work,  and  all  problems 
involving  tact.1 

3.    How  to  Decide  Doubtful  Cases 

How  shall  the  superintendent  decide  quickly  which 
problems  shall  be  statistically  treated  and  which  not? 
Conditions  vary  often  with  each  problem.  But  in 
general,  if  the  big  elements  in  a  given  problem  involve 
numbers  or  can  be  expressed  or  measured  in  numbers, 
statistical  treatment  will  be  applicable.  The  cost  of 
subjects  sure  to  enroll  few  students  cannot  profitably  be 
statistically  compared  with  the  cost  of  subjects  sure 
to  enroll  many  students,  because  the  big  factor  in  the 
situation  cannot  be  stated  in  numbers.  This  factor  is 
the  relative  value  of  the  two  subjects  as  parts  of  educa- 
tion. One  may  correspond  to  salt,  of  which  all  of  us 
need  a  few  grains  each  day,  preferably  some  at  each 
meal,  and  for  which  there  is  no  substitute.  English  is  a 
good  example.  The  other  may  correspond  to  protein, 
which  all  of  us  also  need,  but  which  does  not  have  to  be 
taken  at  every  meal,  or  even  every  day,  and  of  which 
there  are  various  forms.  Science  with  its  numerous 
alternative  forms  is  a  good  example.  Each  subject  is  as 
important  as  the  other  for  perfect  educational  health, 
but  this  relative   importance  can  hardly  be  profitably 

1  Maxwell,  W.  H. :  "How  to  Determine  the  Efficiency  of  a  School 
System,"  American  School  Board  Journal,  May,  1915,  p.  11 


38  School  Statistics  and  Publicity 

treated  or  measured  with  numbers.  Before  rushing 
into  any  statistical  treatment  of  a  school  problem,  the 
superintendent,  then,  should  first  try  to  analyze  out  the 
big  factors,  considering  whether  the  most  important  ones 
can  be  satisfactorily  treated  with  statistical  method. 

EXERCISES 

1.  Which  of  the  following  problems,  or  which  parts  of  them,  may- 
be profitably  subjected  to  statistical  treatment?  Which  may  not? 
Give  your  reasons  precisely  for  each  one. 

(a)  To  what  extent  can  science  be  profitably  taught  in  the 
grades  ? 

(6)  What  percentage  of  the  funds  available  for  library  pur- 
poses in  a  high  school  should  go  to  each  department  in 
a  given  year? 

(c)  Should  home  teachers  be  paid  markedly  lower  salaries  than 

those  from  a  distance? 

(d)  Should  the  passing  mark  be  65  or  70  on  a  scale  of  100? 

(e)  Should  a  given  child  in  a  given  grade  be  promoted? 
(/)    Is  a  given  teacher  marking  too  hard  or  too  laxly? 

(g)  What  salary  as  superintendent  in  a  given  state  may  a  com- 
petent man  reasonably  look  forward  to? 

2.  State  in  the  form  of  definite  questions  at  least  three  school 
problems  in  which  you  are  interested  that  might  profitably  be  subjected 
to  statistical  treatment.     Give  your  specific  reasons  for  so  listing  them. 

3.  Do  the  same  for  at  least  three  school  problems  in  which  you  are 
interested  that  cannot  profitably  be  subjected  to  statistical  treatment. 

II.     HOW    TO    PLAN    STATISTICAL    TREATMENT    OF 
PROBLEMS 

Careful  planning  in  statistical  work  is  always  a  sine 
qua  non  for  success.  "  Each  hour  sp^nt  in  carefully 
arranging  the  work  is  likely  to  save  a  score  of  hours  in 
trying  to  straighten  out  the  confusion  due  to  a  hasty  and 
ill-advised  program."  l  It  is  equally  true  that  "  one 
1  King,  W.  I. :   Elements  of  Statistics,  p.  47 


Collection  of  Data  39 

of  the  peculiarities  of  statistical  work  is  that  practically 
everything  must  be  anticipated  in  advance,  all  possible 
sources  of  error  detected  and  guarded  against,  and  even 
the  general  results  estimated."  ]  The  saving  of  time  in 
statistical  work  becomes  all  the  more  necessary  when  we 
remember  that  the  superintendent  is  at  best  a  very  much 
overworked  man.  Careful  planning  means  time  saved 
for  the  really  big  things  in  the  statistical  work  he  does,  - 
the  results  and  their  meaning  for  his  school  work.  He  is 
also  able  to  get  a  larger  number  of  statistical  processes 
done  in  a  given  time,  which  in  turn  means  that  more 
results  and  meanings  will  be  available.  In  a  word, 
careful  planning  of  statistical  work  permits  a  larger  use 
of  statistical  method  by  the  superintendent  in  the  time 
at  his  disposal. 

To  save  time,  the  main  cautions  to  be  kept  in  mind  in 
planning  statistical  work  will  be  given  briefly  and  dogmat- 
ically with  only  necessary  explanation. 

1.    Decide  Precisely  What  Is  to  Be  Found  Out  or  Proved 
in  the  Statistical  Work 

Indefinite  phrasing  of  the  problem  means  indefinite 
thinking,  with  the  inevitable  wastes  of  time  that  accom- 
pany it.  The  best  device  the  author  has  ever  found  for 
compelling  one  to  make  a  sharp  and  clean-cut  statement 
of  the  problem,  is  to  state  it  in  the  form  of  a  very  definite 
question,  the  adjectives,  adverbs,  subordinate  phrases, 
etc.  of  which  indicate  the  sub-questions  or  minor  problems. 
This  device  has  been  found  to  be  very  serviceable  on  many 
problems  not  involving  statistical  treatment.  It  is 
just   as   serviceable    on   those    that    do    need    statistical 

1  King,  W.  I.  :  Elements:  of  Statistics,  p.  47 


40 


School  Statistics  and  Publicity 


method.  The  usual  topical  statement  of  the  problem  is 
one  of  the  surest  guarantees  of  loose  thinking  at  the 
start.  The  interrogative  form  of  statement  accentuates 
the  problem  effect.  A  superintendent  may  simply  state 
his  problem  as  "  School  Costs  in  Blankville."  How  much 
better  it  would  be  to  state  it  thus :  "  Are  we  paying  all 
we  can  possibly  afford  for  schools  and  are  we  getting  our 
money's  worth?  "  The  same  thing  holds  true  for  sub- 
ordinate problems.  For  example,  compare  the  ordinary 
superintendent's  statement  of  his  problem  with  that  of 
Superintendent  Spaulding  in  the  1912  school  report  for 
Newton,  Massachusetts. 


Usual  Statement 


Report    of     Blankville     Public 

Schools 
Statement  of  aims 


Attendance     and     progress     of 
pupils 


Superintendent  Spaulding's 

Statement 


The  Newton  Schools  :   what  are 
they  trying  to  do? 


Are  they   doing  what  they  are 
trying  to  do? 


(Taken  for  granted) 


Expenditures  of  the  system  for 
the  current  year 


Do  you  approve  of  their  policy? 

Is  their   policy  carried  out  eco- 
nomically? 


Administration 

Course  of  study 

Reports  of  various  supervisors 

Recommendations        for     '  next 
year's  work 


Is  it  administered    efficiently? 


Can  we  afford  to  continue  it? 
Can   we   afford  not    to  continue 

it? 


Collection  of  Data  41 

2.  Plan  to  Collect  only  Data  for  Which  One  Can  Point 

Out  in  Advance  Specific  Ways  in  Which  They 
Will  Be  of  Value  to  Him 

This  does  not  of  course  mean  that  one  can  know  in 
advance  all  the  ways  in  which  the  data  will  be  of  value 
to  him.  The  collection  of  data  that  do  not  seemingly 
answer  some  of  one's  problems  or  promise  to  buttress 
certain  of  his  arguments,  is  simply  a  gambling  proposition. 
And  the  odds  are  ten  to  one  that  the  data  will  never  be 
of  any  material  use  to  him.  It  is  true  that  sometimes 
unexpected  uses  for  data  will  appear  after  the  work  of 
collecting  them  has  begun.  For  example,  the  writer  and 
a  graduate  student  collected  considerable  data  on  the 
cost  of  instruction  in  southern  normal  schools.  But  they 
had  worked  for  some  weeks  before  they  discovered  that 
the  same  data  would  give  material  for  answering  very 
important  questions  about  size  of  classes.  However, 
such  a  valuable  by-product  cannot  be  counted  upon  in 
all  cases. 

3.  Plan    the    Whole    Procedure    Through    to    the    End, 
Trying  It  Out  on  Sample  Data  to  Be  Sure  That  the 

Units,  Blanks,  Processes,  etc.  Will  Work 

Here  is  the  place  where  one  hour  of  good  work  will  save 
twenty  later  on,  as  Professor  King  says.  The  units 
chosen  should  be  carefully  tested  to  see  if  they  are  prac- 
tical. The  blanks  should  be  drawn  out  in  detail  and  the 
actual  operations  attempted  with  them. 

In  particular,  what  is  known  as  the  question  of  "  group- 
ing "  must  be  decided.  This  means  that  if  the  data 
are  to  be  considered  in  groups,  the  exact  range  of  each 
group  must  be   determined   beforehand.     For  example, 


42  School  Statistics  and  Publicity 

if  one  is  studying  days'  attendance,  are  the  children  to  be 
grouped  as  those  attending  0-19,  20-39,  40-59,  etc.,  or 
0-9,  10-19,  20-29,  etc.  ?  This  cannot  be  treated  in  detail 
here,  but  is  discussed  fully  on  pages  106  and  107. 

If  the  blanks  are  to  be  filled  out  by  outside  persons, 
some  of  the  actual  people,  or  preferably  similar  but  less 
intelligent  people,  should  be  used  to  test  out  the  blanks. 
The  errors  that  these  persons  make  should  be  noted  and 
the  blank  revised  accordingly.  Thus,  in  sending  out  a 
blank  for  teachers  to  fill  out,  it  is  advisable  to  submit  the 
rough  draft  to  several  average  teachers,  see  how  they  can 
fill  it  out,  and  revise  as  necessary. 

A  very  helpful  device  at  this  point  is  to  make  a  "  brief  " 
of  just  what  is  to  be  found  and  of  the  methods  to  be  used. 
This  can  be  elaborated  from  the  material  accumulated 
under  the  suggestions  in  i  and  2. 

The  actual  processes  for  handling  the  data  should  be 
tested  on  the  blanks  themselves.  Thus  in  making  out 
a  blank,  if  percentages  are  later  to  be  calculated,  the 
numbers  from  which  they  are  to  be  calculated  should  come 
in  adjacent  columns  if  possible.  Actually  calculating 
such  percentages  on  sample  blanks  will  insure  that  an 
economy  of  this  nature  is  cared  for.  Again,  in  making 
out  blanks  in  series,  the  same  fact  should  appear  if  possible 
in  relatively  the  same  column  in  different  blanks.  This 
will  make  all  manipulation  and  calculation  much  easier. 
But  such  placing  is  almost  sure  to  be  overlooked  unless 
the  calculation  with  simple  data  method  is  carried  through. 

In  all  probability  the  reader  by  this  time  is  asking  this 
question :  "  But  how  can  one  keep  open-minded  if  so 
much  planning  is  to  be  done  ahead?  "  This  is  a  natural 
question.  So  is  the  usual  one  as  to  whether  such  planning 
will  not  tend  to  the  buttressing  of  preconceived  opinions 


Collection  of  Data  43 

rather  than  to  the  discovery  of  anything  new.  In  the 
attempt  to  find  the  truth  about  anything,  the  question 
method  of  outlining  the  plans  is  undoubtedly  the  surest 
mechanical  device  to  aid  in  keeping  one  open-minded. 
Note  that  Superintendent  Spaulding  does  not  suggest 
the  answer  to  any  one  of  his  questions.  Beyond  this, 
probably  no  device  will  be  of  much  service  to  a  person 
who  is  determined  to  prove  a  certain  thing  by  statistics 
whether  or  no. 

Furthermore,  much  of  the  superintendent's  statistical 
work  is  for  the  purpose  of  demonstrating  or  proving  to 
others  what  the  superintendent  already  knows  to  be 
true.  Here  the  element  of  keeping  open-minded  does  not 
enter,  and  planning  ahead  is  unquestionably  very  helpful. 

EXERCISE 

Take  any  one  problem  from  your  list  in  Exercise  2,  page  38.  Out- 
line in  question  form  precisely  what  you  would  wish  to  find  or  prove. 

III.   HOW   TO   DETERMINE   UNITS    AND    SCALES 

In  planning  statistical  work,  the  need  of  units  and 
scales  early  becomes  apparent.1  The  superintendent  is 
constantly  called  upon  to  pass  judgment  upon  the  worth 
of  many  school  matters.  He  generally  does  this  by 
merely  placing  the  thing  judged  in  its  proper  place  in  a 
graduated  scale  of  values  of  such  things.  For  example,  a 
superintendent  passing  judgment  on  the  work  of  a  teacher 
merely  puts  her  in  her  proper  place  in  the  list  of  teachers 
in  his  school  system,  ranged  from  high  to  low,  or  in  a  list 
made  up  of  all  the  teachers  he  has  ever  observed.     This  is 

1  See  pp.  3-11  for  examples  of  errors  arising  from  a  lack  of  proper 
units  and  scales. 


44  School  Statistics  and  Publicity 

indicated  in  the  very  language  he  uses :  "  Best  I  ever 
had,"  "  worst  I  ever  saw,"  "  hopeless,"  "  practically 
perfect,"  etc.  If  he  expresses  his  judgment  in  letters, 
as  A,  B,  etc.,  or  in  figures  as  85,  90,  100,  etc.,  he  is  merely 
substituting  symbols  for  such  word  estimates. 

1.    Subjective  and  Objective  Scales 

It  is  generally  recognized  that  the  superintendent 
will  vary  at  different  times  in  his  judgments  of  the  same 
teacher,  engaged  in  the  same  kind  of  work.  He  may  be 
suffering  from  a  severe  headache,  or  be  perturbed  over  a 
recent  business  reversal.  But  his  readings  of  a  standard 
thermometer,  when  it  was  at  the  same  temperature, 
would  vary  little  from  time  to  time. 

Let  us  now  consider  the  reasons  for  this  difference. 
The  thermometer  is  graduated  into  constant  definite  units 
that  measure  the  same  amount  of  heat  in  the  room  always 
in  the  same  way.  Not  so  in  the  case  of  judging  the 
teacher.  The  superintendent's  scale  of  teaching  ability 
has  no  definite  units  that  always  measure  the  same  amount 
of  teaching  efficiency. 

These  two  scales  represent  the  extremes  of  the  kinds  of 
scales  that  the  superintendent  must  use.  The  units  in 
the  scale  for  judging  teachers  are  in  the  superintendent's 
mind.  Granting  that  he  can  transmit  fairly  clear  ideas 
of  his  scale  to  others,  there  will  be  great  disagreement 
among  those  using  it.  If  they  agree,  they  may  easily 
be  unaware  of  the  fact,  for  the  same  descriptive  words 
mean  very  different  things  to  different  persons.  Since  in 
a  scale  of  this  kind  there  are  no  units  that  can  be  made  to 
mean  exactly  the  same  thing  to  different  people,  such  a 
scale  is  said  to  be  a  "  subjective  "  one.     In  the  case  of 


Collection  of  Data  45 

the  scale  for  measuring  temperature,  the  units  are  not 
concealed  in  the  mind  of  the  person  using  the  scale; 
they  are  external  to  every  one  who  wishes  to  use  it.  For 
this  reason,  such  a  scale  is  called  "  objective."  The 
chances  for  error  or  difference  of  opinion  in  reading 
the  units  of  an  objective  scale  are  slight  as  com- 
pared with  those  arising  from  the  use  of  a  subjective 
one. 

Between  these  two  types  of  scales  lie  others  with  vary- 
ing degrees  of  subjectivity  and  objectivity.  For  example, 
the  Thorndike  handwriting  scale  is  made  up  of  samples 
of  handwriting  rated  from  0  to  18,  as  determined  by  the 
combined  judgment  of  a  considerable  number  of  com- 
petent judges.  This  scale  is  objective  in  that  any  one 
can  see  the  sample  of  handwriting  grades,  say  No.  8. 
But  its  use  is  subjective  in  that  all  people  do  not  agree 
that  No.  8  is  worse  than  No.  9,  nor  would  all  judges  rate 
any  other  sample  of  handwriting  at  the  same  place  on 
the  scale,  say  No.  12.  If  the  number  of  judges  were 
large  enough,  the  variation  in  placing  such  a  sample 
might  range  from  10  to  17. 

It  is  highly  desirable  in  planning  any  statistical  work 
to  try  to  secure  units  and  scales  that  shall  be  as  objective 
as  possible  and  that  shall  have  a  minimum  of  harmful 
subjective  elements. 

Thus,  the  step  between  94  and  95  in  the  marking 
system  of  two  teachers  does  not  mean  at  all  the  same 
thing.  Again,  merely  because  several  people  think  two 
words  are  equal  in  difficulty,  it  does  not  make  them  so. 
Professor  Thorndike,  as"  noted  on  paw  9,  quotes  Rice  as 
counting  that  disappoint  is  equal  to  feather  in  difficulty 
in  spelling,  or  as  proceeding  as  though  it  were.  But  by 
actual  experimentation  in  a  5A  grade,  twenty-four  times 


46  School  Statistics  and  Publicity 

as  many  girls  and  thirteen  times  as  many  boys  missed 
disappoint  as  missed  feather.  In  measuring  arithmetic 
work,  it  is  much  better  to  take  examples  from  the  Courtis 
or  Stone  tests,  because  the  practical  worth  of  these  has  been 
demonstrated  by  the  actual  achievements  of  thousands 
of  pupils. 

One  reason  why  subjective  scales  are  so  often  un- 
desirable is  that  the  zero  points  on  them  are  unknown. 
On  an  objective  scale,  such  as  length,  90  inches  is  just 
three  times  as  long  as  30  inches,  or  it  is  just  three  times 
as  far  from  zero  length.  But  in  the  grading  or  giving  of 
marks  by  two  teachers,  there  is  no  assurance  that  each 
regards  90  as  just  three  times  as  good  as  30  or  just  three 
times  as  far  removed  from  utter  failure.  One  teacher 
on  a  test  may  grade  the  worst  student  in  the  class  at  30 
and  the  best  at  90.  Another  teacher  might  grade  these 
same  students  on  the  same  test  at  60  and  90.  The  teachers 
would  obviously  be  grading  from  different  zero  points. 
In  the  standard  scales  for  grading  composition,  hand- 
writing, and  so  on,  the  zero  points  have  been  determined 
by  a  procedure  too  complicated  to  be  given  here.1  This 
is  why  such  scales,  even  when  they  involve  many  sub- 
jective elements,  are  so  superior  to  the  attempt  of  a 
novice  at  making  his  own  scale. 

If  we  must  take  subjective  estimates  as  units  and  make 
our  own  scale,  it  is  better  to  pursue  the  following  treat- 
ment : 

1.  Avoid  choosing  estimators  with  known  or  probably  marked 
prejudices.  ** 

2.  Have  all  these  persons  estimate  the  worth  of  the  problems  in 
terms  of  a  separate  problem,-  which  for  convenience  is  to  be  consid- 

1  See  Thorndike :  Menial  and  Social  Measurements,  Revised  Edi- 
tion, p.  \6ff. 


Collection  of  Data  47 

ered  worth  so  much,  say  10.     (That  is,  all  call  this  problem  the  value 
of  10.) 

3.  For  the  value  of  any  other  problem,  take  the  average  of  the 
estimates  given  by  the  different  persons.1 

2.    The  Jingle  Fallacy 

The  superintendent  must  beware  especially  of  con- 
sidering things  equal  because  they  are  called  by  the  same 
words.  This  is  known  as  the  "  jingle  "  fallacy.2  Thus, 
one  child  does  not  equal  another  child  as  a  matter  of 
school  expenditure,  if  the  first  child  is  in  the  primary 
grades  and  the  second  child  is  in  the  last  year  of  the  high 
school.  The  cost  of  educating  the  latter  for  one  year  is 
much  more  than  in  the  case  of  the  former.  The  differ- 
ence between  the  ability  to  do  one  problem  and  the 
ability  to  do  two  problems  in  the  Courtis  tests  is  not  the 
same  as  the  difference  in  ability  to  do  fifteen  problems 
and  the  ability  to  do  sixteen  problems.  Any  one  who 
can  do  fifteen  can  fairly  easily  work  up  to  sixteen.  But 
if  a  child  can  barely  do  one,  it  is  a  tremendous  task  to 
work  up  to  doing  two.  The  "  jingle  "  fallacy  usually 
results  from  neglecting  to  define  units  or  to  consider  the 
zero  points. 

3.    Essentials  of  a  Valid  Scale 

The  construction  of  a  good  scale  for  many  lines  cf 
school  work  demands  considerable  technical  knowledge 
and    experience.     The    superintendent    in    general    had 

1  Professor  Thorndikc  on  pages  9  and  10  of  Mental  and  Serial 
Measurements  has  a  much  more  complicated  method  for  utilizing  sub- 
jective estimates. 

2  Professor  Thorndike  borrows  this  term  from  Professor  Aikins, 
See  Mental  and  Social  Measurements,  p.  10. 


48  School  Statistics  and  Publicity 

better  plan  to  use  scales  already  worked  out.1  Beyond 
this  we  may  for  our  purposes  summarize  the  essentials 
of  a  valid  scale  from  Thorndike : 2 

1.  The  scale  must  be  as  objective  as  possible. 

Its  meaning  must  be  such  that  all  competent  judges  will  agree 
as  to  what  it  is. 

2.  The  series  of  facts  used  in  making  up  the  scale  must  be  of  the 

same  sort  of  thing  or  quality. 

3.  The  steps  in  the  scale  should  be  clearly  defined. 

It  is  better  if  they  are  equal ;  if  unequal,  let  the  steps  be  de- 
fined as  definitely  as  possible.  However,  a  scale  in  which 
only  the  order  or  rank  of  the  various  facts  making  it  up  is 
known,  is  often  very  useful. 

4.  The  zero  point  must  be  defined  if  possible. 

4.    Discrete  and  Continuous  Series 

It  is  impossible  to  use  a  scale  properly  unless  one  knows 
whether  the  facts  it  is  to  measure  are  in  a  discrete  or  a 
continuous  series.  A  series  is  said  to  be  discrete  if  it  is 
regarded  as  broken  up,  i.e.,  the  different  items  are  separate 
or  there  are  gaps  between  them.  On  the  other  hand,  if 
the  series  is  capable  of  any  degree  of  subdivision,  that  is, 
if  the  items  are  regarded  as  strung  out  along  the  scale, 
and  running  into  each  other,  the  scale  is  said  to  be  con- 
tinuous. The  table  of  the  costs  of  instruction  in  mathe- 
matics, page  18,  is  an  example  of  a  discrete  scale.  In  this 
table  every  item  is  regarded  as  an  integer  and  there  are 
gaps  between  the  items. 

A  good  example  of  a  continuous  series  i.i^Table  2,  made 
up  from  data  worked  out  by  the  writer. 

1  See  "Descriptive  List  of  Standard  Tests,*'  by  W.  S.  Gray,  Ele- 
mentary School  Journal,    18:56.     (Sept.,  1917.) 

-Thorndike,  E.  L. :    Mental  and  Social  Measurements,  pp.  11-18. 


Collection  of  Data  49 

Table   2.    Continuous  Series  Showing  Fifth  Grade  Achieve- 
ments with  Courtis  Tests  in  Addition,  in  a  Western  City 


Number 

of 

Number  of  children 

problems  attempted 

making  each  score 

1 

0 

2 

2 

3 

10 

4 

24 

5 

32 

6 

35 

7 

34 

8 

54 

9 

25 

10 

27 

11 

24 

12 

4 

13 

4 

14 

3 

15 

6 

16 

9 

17 

2 

18 

2 

In  such  a  series  as  this  one,  the  32  children  who  at- 
tempted five  problems  are  not  all  regarded  as  being  pre- 
cisely at  the  point  "five  problems  attempted"  on  the 
scale,  but  as  distributed  from  "  five  problems  attempted  " 
to  "  six  problems  attempted."  For  this  reason  it  is 
very  important  to  know  what  a  given  number  means 
on  a  scale.  That  is,  does  6  mean  from  5.5  to  6.5,  or  from 
6  to  6.99,  or  nearer  6  than  either  5  or  7?  The  second  of 
these  methods,  that  of  measuring  in  terms  of  the  point 
last  passed,  is  often  the  natural  way  and  saves  labor  in 
allsortsof  measurements.1  This  method  is  the  one  to  use 
where  it  is  possible  to  say  authoritai  ively  that  a  given  case 
1  See  Thorndike:    Merita'  cud  Sarin!  Measurements,  p.  22. 


50  School  Statistics  and  Publicity 

is  beyond  a  certain  point  on  the  scale,  but  "  the  how  much 
beyond  "  cannot  be  easily  determined.  Obviously  it  is  a 
good  method  for  Table  2,  since  a  given  case  has  attempted 
say  five  problems,  but  one  cannot  easily  tell  whether  it  has 
just  started  on  five,  is  half  way  through  the  fifth,  or  is 
practically  ready  to  start  on  the  sixth.  The  other  method 
can  be  used  with  a  scale  like  the  hand-writing  scales,  where 
a  given  case  is  said  to  be  nearer  a  given  sample  on  the 
scale  than  anything  else.  A  case  would  be  called  "9," 
for  example,  without  being  definitely  located  as  either 
below  or  above  that  point.  Here  "  9  "  would  mean  from 
8.5  to  9.5. 


5.    How  to  Use  Scales 

In  actual  practice  the  superintendent  can  measure  the 
worth  of  his  work  in  whole  or  in  part,  on  one  of  three 
kinds  of  scales : 

a.  He  can  place  the  thing  measured  in  its  relative  position 
in  a  scale  of  items  (school  systems,  rooms,  classes,  etc.), 
all  considered  from  the  same  viewpoint  and  without  the 
use  of  units. 

This  is  the  method  used  by  the  superintendent  or  school  board  of 
Blanktown,  when  he  announces  that  his  town  has  the  best  schools 
in  the  state,  or  that  So-and-So  makes  this  statement.  It  of  course 
carries  no  weight  whatever  unless  we  know  that  the  judgment  of  the 
one  making  the  statement  is  sound.  If  one  could  read  all  the  small 
town  papers  of  any  given  state  for  one  year,  he  woTild  probably  find 
three-fourths  of  them  claiming  that  their  home  town  had  the  best 
schools  in  the  state.  The  same  method  with  all  its  weaknesses  is 
often  used  by  a  teacher  in  regard  to  the  value  of  his  pet  method 
of  teaching,  his  favorite  mode  of  discipline,  or  the  particular  class 
he  happens  to  be  teaching  at  this  time. 


Collection  of  Data 


51 


Table  3.     Ranks  of  Certain  Cities  on  Real  Wealth  and 

Assessed  Wealth  behind  Each  $1  Spent  on  Schools 

(Adapted  from  Portland  Survey,  PP.  80,  304) 


City 


Newark,  N.  J. 
Worcester,  Mass. 
Toledo,  O.    .     .     . 
New  Haven,  Conn. 
Paterson,  N.  J. 
Lowell,  Mass.   .     . 
Fall  River,  Mass. 
Syracuse,  N.  Y. 
Cambridge,  Mass. 
Grand  Rapids,  Mich 
Dayton,  O.        .     . 
Washington,  D.  C. 
Scranton,  Pa.    . 
Jersey  City,  N.  J. 
Columbus,  O.    .     . 
Rochester,  N.  Y.  . 
Denver,  Colo.  .     . 
Albany,  N.  Y. 
Providence,  R.  I. 
Bridgeport,  Conn. 
Kansas  City,  Mo. 
Minneapolis,  Minn. 
New  Orleans,  La. 
Louisville,  Ky. 
Nashville,  Tenn.    . 
Omaha,  Neb.     .     . 
Oakland,  Cal.    .     . 
Seattle,  Wash.  .     . 
Spokane,  Wash.     . 
St.  Paul,  Minn.      . 
Indianapolis,  Ind. 
Memphis,  Tenn.    . 
PORTLAND,  ORE. 
Birmingham,  Ala. 
Richmond,  Va. 


Real  Wealfh 

behind  Each 

Rank 

$1  for  Schools 

$165 

1 

180 

2 

184 

3 

185 

4 

185 

5 

194 

6 

196 

7 

202 

8 

204 

9 

207 

10 

208 

11 

212 

12 

216 

13 

218 

14 

221 

15 

225 

16 

231 

17 

234 

18 

256 

19 

276 

20 

280 

21 

294 

22 

314 

23 

326 

24 

350 

25 

352 

26 

354 

27 

364 

28 

370 

29 

407 

30 

408 

31 

449 

32 

456 

33 

479 

34 

536 

35 

Assessed  Wealth 

behind  Each 

$1  for  Schools 


$165 
180 
110 
185 
185 
194 
196 
180 
204 
166 
125 
148 
173 
218 
133 
180 
116 
234 
256 
276 
140 
132 
236 
228 
263 
53 
177 
164 
152 
204 
245 
247 
260 
240 
402 


52  School  Statistics  and  Publicity 

A  refinement  of  the  same  method  is  used  by  a  judge  in  a  contest, 
when  he  ranks  the  contestants  in  order  of  merit  only.  He  then 
gives  the  best  contestant  the  rank  of  1,  the  next  best  the  rank  of  2, 
and  so  on.  But  he  wisely  refrains  from  attempting  to  say  how  much 
better  is  the  first.1 

b.  He  can  compare  his  own  school  with  other  schools  on 
a  scale  of  his  own  making,  all  schools  being  measured  with 
definite  units. 

Thus,  he  may  wish  to  compare  his  community  with  others  on  the 
basis  of  the  amount  of  money  it  really  pays,  wealth  considered,  for 
schools.  The  Portland  Survey  Commission  found  in  comparing  the 
wealth  and  school  expenditures  of  that  city  with  thirty-six  others 
nearest  it  in  size,  that  there  was  a  vast  difference  between  the 
assessed  and  the  real  wealth  in  many  cities.  To  show  this  point 
more  clearly  the  table  on  p.  51  has  been  adapted  from  the  Portland 
Survey. 

It  may  be  seen  from  this  table  that  it  is  very  important  to  have  as 
the  unit,  the  number  of  dollars  of  real  wealth  in  the  city,  not  the 
number  of  dollars  of  assessed  wealth.  If  assessed  valuation  were 
taken  as  the  unit,  Omaha  would  be  1  instead  of  26  ;  Denver  3  instead 
of  17  ;   Lowell  20  instead  of  6,  etc. 

It  is  evident  that  any  one  seeing  this  or  a  similar  scale  must  agree 
to  the  ranking  of  each  city  or  item  as  shown  in  the  first  column, 
provided  the  original  data  for  the  calculation  are  correct  and  the  unit 
is  a  reasonable  one.  Then  there  remains  only  the  question  as  to 
whether  the  cities  selected  for  the  scale  are  representative  ones  for 
fair  comparison  in  the  matter  under  discussion. 

c.  He  may  compare  his  own  school  with  other  schools  by 
means  of  a  standard  test,  and  then  place  his  school  on  a 
scale  of  cities  made  up  as  in  h,  or  merely  compare  it  with 
the  standards  of  the  makers  of  the  test.  {This  amounts  to  a 
scale.) 

The  advantage  of  a  scale  of  this  kind  is  that  the  units  have  been 
proved  equal  or  approximately  equal,  and  there  can  be  no  question 
of  the  relative  positions  of  samples  in  a  given  scale.     And  as  time 

1  See  pp.  11,  192-198 


Collection  of  Data  53 

goes  on,  very  authoritative  scales  will  appear.  Thus,  we  now  have 
scales  of  this  sort  in  the  Thorndike  or  the  Ayres  handwriting  scales, 
the  Hillegas  or  the  Harvard-Newton  composition  scales,  the  Ayres 
spelling  scale,  etc.  Furthermore,  by  taking  the  achievements  of 
school  systems  as  measured  on  these  scales  or  with  standard  units,  a 
superintendent  can  easily  make  a  scale  of  such  achievements  and 
see  where  his  school  system  comes  on  it. 

6.    Practical  Examples  of  Units  and  Scales  for 
Superintendents 

The  best  practical  examples  of  units  and  scales  for  a 
superintendent,  of  course,  appear  in  the  recent  school 
surveys.  A  superintendent  wishing  to  get  up  a  scale 
or  find  units  on  a  given  problem  can  get  them  very  quickly 
by  utilizing  the  following  table.  The  particular  survey 
is  denoted  by  the  name  of  the  city,  and  the  numbers 
refer  to  pages. 

Units  and  Scales  that  a  Superintend ent  may  Profitably  Use 

Description  Where  found  (Surveys) 

Playgrounds 

r  Salt  Lake  City,  222 
Square  feet  per  child  <  Rockford,  7;   Ashland,  11 

{ Denver,  11,  122 
Buildings 

,     .  ..,,  f  Salt  Lake  City,  222 

Square  feet  per  child  i 

\  Ashland,  11 

Cubic  feet  per  child  Ashland,  11 

Space  per  teacher  and  child  Leavenworth,  48 

Number  of  sittings  per  room  Snedden  and  Allen,  29 

Total     seating     capacity     by 

buildings  Snedden  and  Allen,  29 

.  ,.    ,  f  San  Antonio,  315 

Average  cost  per  cubic  foot         <^  _,     .      „  ,  ,    00   00 

1  \  Springfield,  22,  23 

Average  cost  per  pupil  San  Antonio,  315 

Average  cost  per  classroom  San  Antonio,.  315 

Same  for  fuel  Denver,  1,  55 


54 


School  Statistics  and  Publicity 


Same  for  repairs 
Lighting  by  candle  power 

Janitor's  salary  per  room 

Same  per  hour 

Same  per  1000  cubic  feet 

Valuation  per  room 

Teaching  staff 
Number  of  children  per  super- 
visory officer 
Same    on    average    daily    at- 
tendance 

Number  of  children,   average 
daily  attendance  per  teacher 

Training  of  teachers 
Years  of  experience 
Years  of  training 

Teachers'  salaries 


On  yearly  basis 


On  monthly  basis 

On  weekly  basis 

Based  on  enrollment 

Based  on  years  taught 

Maximum  and  minimum  sal- 
aries 

Principal's  salary  based  on 
number  of  rooms  in  building 
Proportionate  expenditures 

Percentages  of  school  expendi- 
tures for  different  purposes 


Denver,  1,  62 

Salt  Lake  City,  235 
J  San  Antonio,  249 
\St.  Louis,  IV,  117-120 

Ashland,  11 

Ashland,  11 

San  Antonio,  250 


Salt  Lake  City,  39 

Oakland,  26 

Salt  Lake  City,  53 

Louisville,  33 

Newton,  1913,  Table  IX 

Oakland,  24 

South  Bend,  198 
South  Bend,  200 

f  South  Bend,  101 

Leavenworth,  50 

Butte,  120 

Newton,  1913,  V 

Janesville,  43,  44 

Bridgeport.  17 

Vermont,  225 
I  Springfield,  61 

Ashland,  14 

Vermont,  225 

Baltimore,  74 

Salt  Lake  City,  55,  56 

Portland,  75  ^ 
f  Cleveland,  97 
\  Springfield,  61 

|  Janesville,  74 

1  St.  Louis,  IV,  55,  56 


Collection  of  Data 


55 


Percentage  of  salaries  for  high 

schools 
Percentage  of  city  expenditures 

for  schools 

Per  capita  costs 
Total  population 

For  each  of  population  over  15 

For  each  adult  male 

High  school  costs  per  person 

in  population 
Average  daily  attendance 

Same  for  fuel 

Enrollment 

Average  number  belonging 
Cost  of  instruction 

Student  hour 


Per  pupil 

Per  pupil  enrolled 

Per  1000  student  hours 

Miscellaneous  costs 

Expenditures  for  whole  cities 

on  medical  inspection 
Evening  schools  per  session 
Per  wagon,  rural  consolidated 

schools 
Part  of  each  $1000  spent  on 

instruction  in  each  subject 


Janesville,  82 

/  Janesville,  68 
\  St.  Louis,  IV,  32 

'  Butte,  143-4 

Bridgeport,  21 
•  Janesville,  68 
I  Oakland,  44 
I  Baltimore,  34 

Portland,  407 

Portland,  407 

Springfield,  95,  96 
In  most  surveys 
Butte,  82 
Oakland,  44 
Kansas  City,  82 
Birmingham,  36 
Houston,  83 
Snedden  and  Allen,  17 

f  South  Bend,  204 

Vermont,  227 

Leavenworth,  51 
[Springfield,  114,  97 
/Rockford,  111 
\  Janesville,  75 

Springfield,  114,  97 

San  Antonio,  215 

Denver,  1,  60 
I  Janesville,  90,  100 


South  Bend,  177 
Newton  (1913),  41 

Texas,  33 

San  Antonio,  213 


56 


School  Statistics  and  Publicity 


Expense  for  attendance  officer 
per  1000  pupils  enrolled 
Time  spent  on  each  subject 

Minutes  per  week 

Part  of  each  1000  hours  spent 
on  each  subject 

Hours    of   recitation    and   di- 
rected study  in  reading  and 
history 
Population 

Number  per   1000   in   certain 
occupations 

Same  by  100's 

Races  by  100's 

Family  for  nearly  everything 

Wealth 

Per  capita  population 


Same  for  real  wealth 

Taxable    wealth    behind    each 
child  in  school 

Same  for  child  in  average  daily 
attendance 

Real   wealth   behind   each   $1- 
spent  on  schools 

Possible  revenue  per  child  en- 
rolled 
Tax  rate 

Mills  on  assessed  valuation 

Same  on  real  valuation 


Same  per  $100  real  valuation 


Portland,  390 

I"  Salt  Lake  City,  76 
'  Leavenworth,  54 
Houston,  83 

San  Antonio,  214 


Cleveland,  121,  125 


South  Bend,  141 
Salt  Lake  City,  17  ' 
Cleveland,  21 
Cleveland,  21 
Red  River,  42,  48,  82 

South  Bend,  211 
Janesville,  68 
St.  Louis,  IV,  18 
Salt  Lake  City,  19 
Cleveland,  25 
Oakland,  43 
Maryland,  128 

Janesville,  52 

Portland,  414 

Salt  Lake  City,  48 

Illinois,  262^ 
/Portland,  108 
\  Leavenworth,  17 
[South  Bend,  215 
j  Janesville,  60 
j  Salt  Lake  City,  313 
[Rock ford,  119,  49 


Collection  of  Data  57 

Per  $100,000  real  wealth  Cleveland,  25 

Rate  necessary  to  produce  esti- 
mated per  capita  support  for 

schools  on  actual  wealth  Salt  Lake  City,  311 

Enrollment 

Increase  in  children  per  week 

(5  years)  Salt  Lake  City,  36 

On  basis  of  1000  children  in 

kindergarten  Ogden,  9 

Parts  of  100  pupils  in  public, 
private,  and  parochial 
schools  Cleveland,  28 

Same,  failures,  and  promotions       Denver,  1,  70,  2,  77 

EXERCISES 

1.  Discuss  the  value  for  the  superintendent  of  the  units  used  in 
each  of  the  examples  given  on  pages  53  57,  and  of  the  scales  that 
could  be  made  up  from  such  units.  Just  how  would  you  make  up 
these  scales? 

2.  What  is  the  value  of  the  question  for  measuring  the  efficiency  of 
teachers,  counting  the  number  of  questions  they  ask  in  a  given  time? 
Why? 

3.  Which  is  better  for  measuring  the  preparation  of  high  school 
teachers,  and  why? 

(a)  The  number  cf  years  they  were  in  college. 

(b)  The  number  of  years  beyond  the  elementary  school  they 

spent  in  study. 

4.  Precisely  what  is  the  value  of  each  of  the  following  methods  of 
instructing  judges  in  a  contest,  and  why? 

(a)  Mark  on  a  scale  of  100. 

(b)  Mark  on  a  scale  of  30. 

(c)  Mark  the  best  1,  the  next  best  2,  etc. 

(d)  Mark  the  best   100,  the  worst  70,  and   the  others  where 

they  should  come  in  between. 

(e)  Mark  on  a  scale  of  100,  allowing  50  for  content  and  organ- 

ization, 30  for  English,  and  20  for  delivery. 

5.  What  units  and  scales  would  you  plan  to  use  in  studying  the 
statistical  problem  selected  in  the  exercise  on  page  43,  and  just  why? 
Precisely  how  would  you  plan  to  use  the  scale  or  scales  chosen? 


58  School  Statistics  and  Publicity 

IV.    HOW  TO  DO  THE  ACTUAL  COLLECTING 
1.   Records  in  One's  Own  School 

Most  of  the  superintendent's  data  must  come  from  his 
own  school  system.  But  on  many  problems  the  matter 
of  giving  advice  about  the  collecting  may  be  like  that  of 
Holmes,  when  he .  said  that  one  should  always  exercise 
great  care  in  the  selection  of  one's  grandfather.  The 
superintendent  can  at  any  rate  collect  far  more  data  by 
seeing  that  his  records  are  so  kept  as  to  show  the  desired 
facts  later,  than  he  can  ever  suddenly  exact  from  teachers 
who  have  never  thought  of  keeping  or  giving  out  infor- 
mation on  this  point.  For  example,  the  disputes  and 
troubles  in  the  studies  of  retardation  and  elimination  a 
few  years  ago  arose  mainly  because  of  the  way  school 
records  had  been  kept.  It  was  impossible  to  tell  from 
existing  records  how  many  of  the  given  pupils  had  entered 
the  first  grade  at  any  given  time  years  before,  or  to  find 
all  the  significant  facts  about  a  pupil  brought  together 
in  one  place. 

Inasmuch  as  the  facts  in  the  superintendent's  own  school 
are  often  meaningless  unless  they  can  be  compared  with 
similar  facts  in  other  school  systems,  he  must  as  far  as 
possible  use  records  similar  to  those  of  his  fellow  school 
men.  Therefore,  it  is  advisable  that  he  use  the  records 
and  reports  recommended  by  the  Committee  of  the 
National  Education  Association  on  Records  and  Reports, 
and  by  the  United  States  Bureau  of  Education.  He 
should  also  try  to  get  the  State  Department  of  Education 
in  his  state  to  use  blanks  that  will  fit  in  with  this  system. 
If  a  decalogue  for  superintendents  should  be  written, 
one  of  the  first  commandments  undoubtedly  should  be : 
"Thou  shalt  keep  thy  records  as  nearly  as  possible  by 


Collection  of  Data  59 

the  uniform  system  of  the  National  Education  Associa- 
tion." » 

2.    Other  Sources  of  Data 

The  data  from  other  school  systems  are  obtained 
usually  by  the  use  of  questionnaires,  from  printed  reports, 
school  surveys,  magazine  articles,  etc.  Aside  from  the 
question  of  selection,  the  questionnaire  method  is  often 
practically  worthless  for  collecting  statistical  data. 
School  men  are  too  busy  to  answer  large  numbers  of 
questions,  to  work  out  the  object  of  the  questionnaire 
when  this  is  not  clearly  stated,  to  hunt  up  much  infor- 
mation on  former  conditions,  to  puzzle  out  involved  or 
ambiguous  questions,  and  sometimes  too  careless  to  give 
information  definite  enough  to  be  of  any  service.  As  a 
result,  few  replies  will  come  in  on  a  given  problem, 
and  not  all  of  these  will  be  complete.  This  means  few 
opportunities  for  comparison. 

It  is  always  better  to  make  use  of  printed  statistics 
where  possible,  taking  care  to  be  sure  of  the  units  and 
processes  used  in  compiling  the  statistics  taken.  The 
mere  fact  that  they  are  part  of  a  printed  report  or  formal 
presentation,  often  required  by  law,  practically  insures 
much  more  accurate  figures  than  can  be  secured  by  the 
questionnaire  method.  A  second  advantage  is  that  the 
superintendent  can  see  that  all  his  figures  are  on  the  same 
basis,  a  thing  impossible  with  a  questionnaire  because  of 
the  inability  of  many  people  to  understand  or  follow 
directions  on  an  inquiry  which  they  have  never  seen 
before  and  never  expect  to  see  again. 

1  See  Bulletin  II.  S.  Bureau  Education,  1912,  No.  3,  or  get  particu- 
lars from  such  companies  as  the  Library  Bureau  or.  the  Shaw-Walker 
Co. 


60  School  Statistics  and  Publicity 

For  busy  superintendents  the  following  suggestions  as 
to  where  material  for  statistical  comparisons  may  be 
found,  are  appended : 

1.  Report  of  the  United  States  Commissioner  of  Education,  Volume 
II,  each  year  contains  numerous  tables  on  city  school  systems.  It 
includes  such  material  as  enrollment,  number  of  teachers,  aggregate 
days'  attendance,  average  daily  attendance  for  both  elementary  and 
secondary  schools,  length  of  session,  number  of  children  of  census  age 
in  private  schools,  number  of  buildings,  number  of  sittings,  itemized 
receipts  and  expenditures  of  all  school  systems  in  cities  of  5000  or  more. 

2.  The  bulletins  of , the  United  States  Bureau  of  Education  often 
contain  much  valuable  material  on  special  topics.  Especially  good 
are  such  ones  as  the  following : 

A  comparative  study  of  the  salaries  of  teachers  and  officers. 
1915,  No.  31. 

Ayres,  L.  P. :  Provision  for  exceptional  children  in  the  pub- 
lic schools.     1911,  No.  14. 

Boykin,  Jas.  C.  and  King,  Roberta:  The  tangible  rewards 
of  teaching.     1914,  No.  16. 

Deffenbaugh,  W.  S. :  School  administration  in  the  smaller 
cities.   1915,  No.  44. 

Frost,  Norman :  A  statistical  study  of  the  public  school  sys- 
tems of  the  southern  Appalachian  Mountains.     1915,  No.  11. 

Monahan,  A.  C.  and  Dye,  C.  H.  :  A  comparison  of  the  salaries 
of  rural  and  urban  superintendents  of  schools.     1917,  No.  33. 

Morse,  H.  N. :  Educational  survey  of  Montgomery  County,  Md. 
1913,  No.  32. 

Public,  society,  and  school  libraries.     1915,  No.  25. 

Statistics  of  certain  manual  training,  agricultural,  and  industrial 
schools.     1915,  No.  19. 

Strayer,  G.  D.  :    Age  and  grade  census  of  school*  and  colleges. 

1911,  No.  5.  «» 

Thorndike,  fi.  L. :  The  elimination  of  pupils  from  school. 
1907,  No.  4. 

Thorndike,  E.  L. :  The  teaching  staff  of  secondary  schools  in 
the  U.  S.       1909,  No.  4. 

Updegraff,  Harlan  :   A  study  of  expenses  of  city  school  systems. 

1912,  No.  5. 


Collection  of  Data  61 

Updegraff,  Harlan :  Public  and  private  high  schools.  1912, 
No.  22. 

Updegraff,  Harlan  and  Hood,  Wm. :  Urban  amd  rural  school 
statistics.   1912,  No.  21. 

3.  The  reports  of  the  State  Department  of  Education  within  the 
state  where  the  superintendent  lives. 

4.  Strayer,  G.  D.  and  Thorndike,  E.  L.  Studies  in  Educational 
Administration.  (Contains  the  cream  of  many  dissertations  pub- 
lished at  Teachers  College  prior  to  1912.) 

5.  Publications  of  the  United  States  Census  Bureau,  especially 
special  reports  on  cities  and  abstracts  of  each  census. 

6.  School  surveys  of  all  classes. 

7.  Bulletins  issued  at  various  times  such  as  those  started  by  Super- 
intendent Spaulding  at  Minneapolis  in  September,  1916. 

8.  Dissertations  from  leading  schools  of  education  in  universities. 
The  publications  of  the  superintendent's  own  state  university  in  this 
line  will  be  easily  available.  In  addition,  the  dissertations  and  theses 
from  Teachers  College,  Columbia  University,  and  the  School  of  Edu- 
cation at  the  University  of  Chicago,  are  very  valuable. 

9.  Reports  of  investigations  appearing  in  standard  educational 
magazines.     The  best  for  this  purpose  are : 

American  School  Board  Journal,  Milwaukee 
Educational  Administration  and  Supervision,  Baltimore 
Elementary  School  Journal,  University  of  Chicago  Press 
Journal  of  Educational  Psychology,  Baltimore 
School  Review,  University  of  Chicago  Press 
School  and  Society,  New  York 

10.  After  the  foregoing  was  written,  Professor  H.  O.  Rugg's  book 
on  Statistical  Methods  Applied  to  Education  appeared.  It  has  much 
more  extended  references,  especially  on  pages  28-39,  361-375.  The 
material  is  admirably  classified  for  ready  reference. 

EXERCISE 
For  the  special  problem  you  have  selected,  jot  down  as  carefully 
as  you  can  at  this  time : 

(a)  The  places  in  your  own  school  system  (or  one  in  which  you 

are  interested)  from  which  you  might  secure  statistical 
data  on  it. 

(b)  Other  likely  sources  of  statistical  data  on  it. 


62  School  Statistics  and  Publicity 

3.    Sampling 1 

As  soon  as  the  sources  of  data  are  determined,  the 
question  arises  as  to  what  can  be  done  in  the  way  of 
"  sampling  "  with  a  view  to  cutting  down  the  inevitably 
large  amount  of  work.  "  Sampling,"  of  course,  means 
working  from  selected  or  typical  specimens  rather  than 
with  the  whole  mass  of  data. 

The  superintendent  usually  considers  sampling  be- 
cause he  must  answer  one  of  three  questions: 

1.  How  many  measures  are  needed  of  an  item  to  be  sure  that  the 
item  is  fairly  well  represented? 

2.  How  many  cases  and  which  ones  need  to  be  treated  in  a  large 
mass  of  data  to  be  sure  that  the  results  will  be  approximately  true  of 
the  whole? 

3.  In  case  the  superintendent  at  best  can  get  only  a  small  list  of 
items  for  comparative  purposes  (say  only  a  dozen  towns  that  are 
really  comparable  to  his  town  on  school  expenditures),  how  is  he  to 
choose  these  items-? 

Let  us  now  take  these  up  in  order. 

Number  of  Measures  of  One  Item  Needed.  No 
arbitrary  rule  can  be  laid  down  for  the  number  of  measures 
needed  on  any  one  item.  But  it  is  safe  to  say  that  often 
there  should  be  more  than  one  in  order  to  insure  a  reliable 
average  measure  for  the  item.  For  example,  in  comparing 
cities  for  school  expenditures,  it  is  often  very  unfair  to 
get  the  expenditures  for  one  year  only.  There  may  be 
very  unusual  conditions  for  that  year  \n  several  cities, 
say  fires,  cold  spells  that  sent  up  fuel  bills,  epidemics  that 
necessitated  much  medical  expense,  etc.  In  such  studies 
it  is  customary  to  take  the  average  of  two  years  for  each 
city,  as   Strayer  does   in   his   City   School   Expenditures. 

1  See  also  pp.  22-23 


Collection  of  Data  63 

Superintendent  Spaulding,  in  a  recent  monograph  on 
expenditures  in  Minneapolis,  takes  the  average  for  five 
years  for  all  cities  studied.  Another  good  example  is 
seen  in  comparing  teachers  on  their  ways  of  marking 
students.  It  is  very  unfair  to  judge  a  teacher  by  the 
marks  she  gives  at  any  one  term  examination,  or  in  any 
one  class.  Professor  Max  Meyer  at  the  University  of 
Missouri,  for  instance,  never  attempted  to  pass  judgment 
on  the  marking  of  a  member  of  the  faculty  there  until  he 
could  get  at  least  five  hundred  marks  given  by  that  teacher 
in  all  his  classes.  A  teacher  may  rightly  object  to  a 
rating  given  her  by  the  superintendent  on  one  short 
visitation  in  one  subject  only.  In  practice,  of  course,  she 
will  not  object  to  such  rating  if  it  is  highly  favorable 
to  her,  but  it  is  probably  about  as  far  from  the  exact  truth 
as  an  unfavorable  rating  would  be  if  made  on  the  same 
visitation.  The  writer  some  years  ago,  in  inspecting  high 
schools  for  the  University  of  Missouri,  found  that  the 
inspection  was  hardest  in  this  one  particular.  It  was  im- 
possible to  visit  a  school  oftener  than  once  a  year  for  a 
day,  and  this  made  it  extremely  difficult  to  pass  judgment 
on  four  or  five  teachers  in  the  course  of  the  six  hours  or 
less  of  teaching,  judgment  to  which  the  teachers  or  their 
superintendents  would  really  subscribe. 

The  more  measures  we  take  of  an  item,  provided  they 
are  not  all  chosen  with  the  same  bias  or  cause  for  error, 
the  more  reliable  will  be  the  final  average  measure  taken 
for  that  item.  But  time  is  too  valuable  to  permit  going 
on  indefinitely  getting  measures  of  one  item.  The  safe 
procedure  is  to  take  as  few  measures  of  it  as  may  reason- 
ably be  expected  to  represent  it  fairly. 

Selection  of  Samples  Ordinarily.  The  amount  of 
work  in  any  statistical  treatment  is  so  great  that  the 


64  School  Statistics  and  Publicity 

question  of  cutting  down  the  number  of  items  by  sampling 
is  very  important.  But  it  is  equally  important  that  the 
samples  be  so  selected  as  really  to  represent  the  whole. 
Neglect  of  this  is  responsible  for  the  worthlessness  of 
many  laborious  pieces  of  school  statistics. 

The  selection  of  samples  should  be  absolutely  at  random, 
and  if  there  are  groups  of  data,  the  same  percentage  of 
samples  should  be  taken  from'  each  group.  Thus,  a  su- 
perintendent who  was  making  a  study  of  outside  reading 
done  by  students  in  his  town,  could  not  get  very  trust- 
worthy results  by  asking  the  pupils  in  five  wealthy  families, 
in  five  families  in  comfortable  circumstances,  and  in  five 
poor  families.  There  would  be  so  many  more  in  the 
second  group  than  in  the  first,  and  so  very  many  more  in 
the  third  than  in  either  of  the  other  two  groups  that  his 
results  would  be  untrustworthy.  To  get  at  anything  like 
the  truth,  he  would  have  to  take,  say,  twenty  from  the 
second  group  and  possibly  a  hundred  from  the  third.  It  is 
equally  erroneous  to  attempt  to  obtain  results  on  the  effect 
of  negro  population  on  schools  in  Texas  by  ranging  the 
counties  in  order  of  percentage  of  negro  population  and 
then  taking  three  of  the  counties  most  free  from  negroes, 
three  from  the  middle  group,  and  three  of  those  having 
the  largest  numbers  of  negroes.  The  results  are  wholly 
unreliable  until  we  know  how  many  counties  are  in  the 
lowest  natural  group  (that  is,  one  without  gaps  in  the 
percentages),  how  many  in  the  middle  group,  and  how 
many  in  the  highest  group,  and  that  we  have  taken  the 
same  proportion  of  samples  from  each.  Again,  a  teacher 
in  giving  a  grade  to  a  pupil  ought  not  to  take  the  work  of 
the  pupil  during  the  last  few  days  of  the  term.  It  is  not 
a  fair  sampling  of  the  student's  work.  Nor  should 
she  take  the  sample  grades  for  her  private  book  at  any 


Collection  of  Data  65 

stated  time  known  to  students.  If  they  know  she  always 
grades  them  on  Friday,  they  will  do  well  on  that  day  and 
slow  down  on  other  days. 

Unfair  selection  explains  why  questionnaire  methods 
of  getting  results  are  often  so  unsatisfactory  and  mis- 
leading. The  people  who  answer  questionnaires  are 
often  very  selected  and  biased,  and  do  not  represent  the 
whole  group  at  all  fairly.  For  example,  a  superintendent 
may  wish  to  know  what  parents  think  of  his  school  and 
may  send  out  a  questionnaire  for  this  purpose.  It  will 
be  sure  to  be  answered  mainly  by  those  who  are  favorable 
or  who  wish  to  make  him  think  they  are  favorable,  and 
possibly  by  a  very  few  opponents  who  are  stirred  up 
enough  to  come  out  frankly,  but  who  are  probably  se- 
riously or  unfairly  prejudiced  against  him.  If  he  sends 
out  a  questionnaire  to  other  superintendents,  as  a  general 
rule  only  the  superintendents  who  think  they  can  make 
a  better  showing  than  he  can,  will  send  in  results.  So 
very  serious  is  this  defect  of  the  questionnaire  method 
that  the  phrase  of  President  Kirk  of  the  Kirksville, 
Missouri,  Normal  School,  "  a  questionable  question- 
naire," is  often  justified.  The  most  casual  examination 
of  the  statistical  outpourings  of  the  questionnaire  type 
in  the  last  few  years  will  show  that  very  often  only  a  small 
percentage  of  the  persons  receiving  questionnaires  ever 
answer  them.  This  small  percentage  is  almost  certain  to 
be  selected  on  some  peculiar  bias  and  is  getting  smaller 
because  so  many  foolish  and  needless  questionnaires  have 
been  sent  out  to  superintendents  in  the  last  few  years. 
Many  school  men  confess  to  throwing  nearly  all  question- 
naires into  the  waste  basket  or  turning  them  over  to 
clerks  or  pupils  to  answer. 

A  good  method  of  sampling  is  the  familiar  one  of  taking 


66  School  Statistics  and  Publicity 

every  fourth  or  fifth  case  of  the  items  arranged  alphabeti- 
cally, or  in  order  of  magnitude,  so  that  there  can  be  no 
prejudice  in  the  matter.  This,  of  course,  insures  getting 
the  same  percentage  in  each  of  any  possible  groups.  Thus, 
if  it  is  a  case  of  consulting  citizens,  every  fifth  name  in 
the  telephone  book  would  give  the  superintendent  a  good 
sampling  of  men  able  to  have  telephones.  But  it  would 
not  represent  all  citizens.  To  get  all  represented,  he 
had  better  take  every  fifth  name  in  a  directory  of  the 
city,  or  in  a  list  of  the  registered  voters.  If  it  is  a  case  of 
children,  arranging  them  alphabetically  by  grades,  per- 
haps boys  and  girls  separately,  and  then  taking  every 
fifth  name  would  give  him  a  good  set  of  samples  for  boys 
and  another  good  set  for  girls.  If  a  teacher  has  kept  daily 
grades  of  pupils,  he  can  get  a  good  sample  set  of  grades 
by  taking,  say,  every  seventh  grade  for  each  pupil,  or 
something  of  this  sort,  just  so  long  as  the  grade  taken 
does  not  fall  on  the. same  day  of  the  week  every  time. 

Sampling  is  often  resorted  to  in  giving  standard  tests 
to  large  numbers  of  children  where  the  labor  of  grading 
the  papers  from  all  would  be  very  great.  One  of  the  best 
sampling  schemes  coming  to  the  writer's  notice  is  the  one 
employed  in  the  San  Francisco  survey  in  1916.  Four 
tests  —  arithmetic,  spelling,  penmanship,  and  reading  — 
were  given.  Two  classrooms  in  different  grades  in  the 
eighty-one  elementary  schools  in  the  city  were  chosen  at 
random  for  a  test  in  some  one  of  the  four  subjects.  No 
teacher  or  principal  knew  in  advance  what  rooms  had  been 
chosen  or  what  subject  would  be  given  in  the  room  selected. 

If  a  questionnaire  has  to  be  used,  it  is  generally  advisable 
for  the  superintendent  to  pick  out  a  reasonably  small  list 
by  one  of  the  preceding:  methods,  and  then  to  devote  his 
energies  to  seeing  that  approximately  every  name  on  the 


Collection  of  Data  67 

list  sends  in  a  fairly  accurate  answer.  The  "  personal 
questionnaire  "  filled  out  in  person  by  an  "  interviewer  " 
is  sometimes  an  excellent  device.  It  relieves  the  person 
interviewed  of  much  drudgery  and  insures  correct  inter- 
pretation of  the  questions.  But  it  tends  to  embarrass 
the  person  giving  the  information,  often  to  the  point  of 
stopping  easy  conversation.  To  avoid  this,  the  inter- 
viewer is  apt  to  encourage  or  accept  rough  estimates  in 
place  of  accurate  data.  In  any  case,  the  best  way  for  the 
superintendent  to  get  answers  is  first  to  be  sure  that  his 
investigation  will  be  of  value  to  some  one  beside  himself, 
and  especially  to  those  he  requests  to  fill  out  his  question- 
naire. Then  he  should  promise  to  give  all  those  answering 
it  a  copy  of  the  results.  If  he  ever  intends  to  do  any  more 
investigating,  he  must  faithfully  keep  this  promise, 
aside  from  his  moral  obligation  to  keep  it.1 

Selection  of  Very  Few  Samples.  Often  the  superintend- 
ent knows  that  he  cannot  by  any  possibility  get  more  than 
a  small  number  of  items.  His  problem  then  is  how  to 
select  these  items  so  that  he  may  be  sure  that  his  com- 
parisons will  not  be  absurd.  For  example,  if  he  wishes 
to  know  whether  his  community  is  spending  enough 
money  on  its  schools,  is  he  to  take  all  cities  of  that  size 
in  the  United  States  or  in  his  state  ?  Neither  procedure 
will  do  because  the  same  conditions  manifestly  do  not 
obtain  in  all  of  these  cities.  The  same  would  be  true  if 
the  cities  were  selected  on  the  basis  of  the  number  of 
children  of  school  age,  the  number  of  children  in  school, 
etc.     The   towns   certainly   ought  not   to   be   compared 

1  For  those  who,  in  spite  of  the  preceding,  find  it  necessary  to  use 
a  questionnaire,  pp.  40-56  of  Rugg's  Statistical  Methods  Applied  to 
Education  will  be  very  valuable.  This  contains  the  most  practical 
treatment  of  the  questionnaire  that  the  author  has  yet  seen. 


68 


School  Statistics  and  Publicity 


mercilessly  unless  their  wealth  per  capita  is  something 
like  the  same.  They  cannot  be  compared  on  current 
school  expenditures  unless  we  know  that  their  school 
debts  are  something  like  the  same  relatively,  and  even 
this  may  not  be  enough  We  may  have  to  know  their 
present  taxes  for  other  needs,  the  city's  indebtedness  for 
other  purposes,  etc. 

Mr.  F.  0.  Seymour  in  writing  his  master's  thesis  at 
Peabody  in  1916  had  the  same  problems  in  making  a 
study  of  school  costs  for  Amarillo,  Texas.  Table  4  is 
adapted  from  his  work. 

Table   4.     Showing  School  Statistics  for  Certain  Cities  op 
about  the  same  population  and  general  situation 


Town 

Popu- 
lation 

Per 
cent  of 
pupils 
to  total 
popu- 

Per 
cent  of 
pupils 

in 
school 

Wealth 

per 
capita 

City 

debt 

per 

capita 

City 

tax 

per 

capita 

Per  cent 
of  city 
revenue 

going  to 
schools 

lation 

86 

Coffeyville,  Kan. 

13,687 

24.0 

$635 

$62 

$30 

48.1 

Dennison,  Tex. 

12,632 

19.4 

69 

501 

22 

14 

41.9 

Pearsons,  Kan. 

12,363 

18.3 

71 

815 

31 

15 

52.9 

Sherman,  Tex. 

12,412 

20.9 

71 

608 

20 

15 

39.1 

Guthrie,  Okla. 

11,654 

20.1 

.  72 

474 

68 

14 

34.2 

Marshall,  Tex. 

11,452 

19.4 

61 

409 

51 

13 

33.4 

Paris,  Tex. 

11,269 

27.2 

85 

815 

51 

19 

36.5 

Palestine,  Tex.    ' 

10,480 

18.1 

59 

508 

19 

10 

40.3 

Cleburne,  Tex. 

10,464 

24.3 

78 

508 

31 

14 

40.8 

San  Angelo,  Tex. 

10,321 

15.2 

49 

566 

20 

10 

44.8 

Amarillo,  Tex. 

9,957 

16.8 

61 

549 

«36 

9 

45.7 

It  may  be  noted  concerning  these  cities  which  Mr.  Sey- 
mour has  selected  for  purposes  of  comparison  with 
Amarillo  that :  (1)  They  are  of  practically  the  same 
population  ;    (2)  they  are  all  in  the  same  section  of  the 


Collection  of  Data 


69 


country ;  (3)  the  various  items  given  in  the  table  are  on 
the  whole  rather  close  together  for  the  different  cities. 
Had  he  included  some  eastern,  northern,  or  extreme 
southern  cities  of  the  same  population,  undoubtedly  the 
variations  in  these  items  would  have  been  much  greater. 
Examples  of  Bad  Sampling.  Failure  to  pick  out 
samples  that  really  represent  the  whole  group  may  lead 
to  some  very  fallacious  conclusions.  For  instance, 
Professor  Bobbitt  in  his  article  on  the  cost  of  instruction 
in  high  schools,  for  purely  illustrative  purposes,  compares 
results  from  various  cities.1  The  population  of  the  cities 
he  uses  varies  greatly.  Figures  could  not  be  obtained 
for  all,  but  Table  5  gives  enough  to  show  the  variations. 

Table  5.    Variations  in  Population  of  the  Cities  Used  by 
Professor  Bobbitt  in  His  Study  of  High  School  Costs 


City 

Population 
in  1910 

City 

\Populatio7i 
1    in  1910 

Mishawaka,  Ind. 

Elgin,  111 

Maple   Lake,    Minn. 
Granite  City,  111. 
East  Chicago,  111. 
De  Kalb,  111.    .     .     . 
San  Antonio,  Tex.     . 
Harvey,  111.      .     .     . 
Waukegan,  111.      .     . 
South  Bend,  Ind. 
East  Aurora,  111.  .     . 
Rockford,  111.        .     . 

11,885 
25,976 

9,903 

1 

8,102 
96,614 

7,227 
16,069 
53,684 
29,807  - 
45,401 

Booneville,  Mo.  . 

Brazil,  Ind 

Leavenworth,  Kan. . 
Greensburg,  Ind.      .     . 
Morgan  Park,  111.     .     . 
Noblesville,  Ind.      .     . 
Norfolk,  Neb.      .     . 
Washington,  Mo. 
Bonner  Springs,  Kan.  . 
Russell,  Kan.       .     .     . 
Junction  City,  Kan. 
Mt.  Carroll,  111.  .     .     . 

4,252 
9,340 
19,363 
5,420 
3,694 
5,073 
6,025 
3,670 

5,598 
1,759 

! 

bobbitt,  J.  F. :  "Cost  of  Instruction  in  High  Schools,"  School 
Review,  23:  505-534.      (Oct.,  1915) 

2  Population  for  Aurora.  There  are  two  high  schools,  one  in  East 
Aurora  and  one  in  West  Aurora. 


70 


School  Statistics  and  Publicity 


But  Professor  Updegraff  has  shown  that  the  cost  of 
instruction  is  higher  in  large  cities  than  in  small  ones. 
See  Table  6. 

Table  6.    Variations  in  Cost  of  Instruction  and  Supervision 
per  Capita  (Population)  in  Cities  of  Different  Sizes  1 


Per  capita 

Per  capita 

expenses  of 

Median 
city  in 

Population 

expenses  of 
teachers' 
salaries 

salaries  and 
expenditures 

for 
supervisors 

Total 

Group  I 

300,000  and  over 

$50.98 

$1.18 

$52.16 

Group  II 

100,000  to  200,000 

36.15 

3.26 

39.41 

Group  III 

50,000  to  100,000 

36.93 

3.39 

40.32 

Group  IV 

80,000  to    50,000 

29.25 

3.38 

32.63 

As  to  costs  in  schools  in  cities  below  30,000,  no  such  figures 
are  known  to  the  author.  It  is  entirely  probable  that  they 
are  higher  than  the  lowest  figures  cited  here  because  of  not 
having  such  full  classes  in  high  school.  But  the  main 
point  is  that  there  are  always  great  dangers  in  selecting 
samples  that  cannot  reasonably  be  regarded  as  coming 
from  the  same  class.  Professor  Bobbitt  was,  of  course, 
perfectly  aware  of  this  difficulty  and  on  page  506  of  his 
article  indicates  that  his  tables  are  valuable  as  patterns 
of  work  mainly. 

Another  example  of  the  results  of  unwise  selection 
of  samples  is  furnished  by  Superintendent  Spaulding's 
monograph    on    the    cost    of    the    Minneapolis    schools.2 

1  Adapted  from  Updegraff,  Harlan  :  "A  Study  of  Expenses  in  City 
School  Systems,"  Bulletin  U.  S.  Bureau  Education,  1912,  No.  5, 
pp.  7,  86.      'Computed  on  enrollment  of  pupils) 

2  Sptiulding,  F.  E. :    Financing  the  Minneapolis   Schools.      (Board 

of  Education,  Minneapolis,  Sept.,  1!  ~ ',' > 


Collection  of  Data  71 

He  has  a  chart  on  page  46  showing  the  expenditure  per 
child  for  ordinary  maintenance  of  the  elementary  schools. 
But  in  this  chart  he  has  included  three  southern  cities, 
Louisville,  Birmingham,  and  New  Orleans.  All  three 
have  in  their  population  large  negro  elements  which 
seldom  pay  taxes  and  add  little  wealth  to  the  community 
because  they  are  not  able  to  do  so.  At  the  same  time 
they  have  relatively  large  numbers  of  children  to  be 
provided  with  schooling.  Consequently  these  southern 
cities  have  to  take  care  of  a  large  number  of  negro  children 
with  no  corresponding  increase  in  revenue.  Therefore, 
they  should  not  be  placed  in  comparison  with  Minneapolis. 
Their  presence  in  this  table  has  the  effect  of  making 
Minneapolis  appear  more  extravagant  in  her  expenditures 
for  schools  than  would  be  the  case  if  only  northern  cities 
were  considered. 

Summary  of  Rules  for  Sampling.  These  may  be 
given  briefly : 

1.  Be  sure  that  the  measures  of  any  one  item  represent  its  usual 
state. 

2.  Select  samples  absolutely  at  random. 

3.  If  there  are  groups  of  data,  take  the  same  percentage  of  samples 
from  each  group. 

4.  Avoid  using  a  questionnaire  if  possible.  If  it  must  be  used,  be 
careful  to  discount  the  results  for  classes  liable  to  be  selected  by  it. 

5.  If  only  a  small  number  of  samples  are  obtainable,  select  them 
with  unusual  care. 

4.    Blanks  and  Tabulating 

Devices  for  Blanks.  The  actual  collection  of  data 
should  be  made  on  some  form  for  tabulating,  called  for 
convenience  a  "  blank."  Rules  for  making  blanks  which 
hold  good  for  many  specific  problems  are  extremely  hard 
to  lay  down,  but  the  following  devices  are  often  helpful : 


72 


School  Statistics  and  Publicity 


a.  Plan  the  blanks  so  as  to  get  a  maximum  amount  of 
information  with  a  minimum  of  space. 

This  reduces  chances  for  error,  because  the  more  nearly  a  person 
can  see  all  of  the  blank  at  one  look,  the  greater  mastery  of  it  and 
of  the  relations  of  its  parts  he  will  have.  But  this  is  not  to  be  inter- 
preted as  advocating  extreme  condensation  resulting  in  eye  strain,  or 
the  elimination  of  all  space  for  calculations.  Nor  does  it  mean  that 
setting  all  the  material  on  one  sheet  of  paper  is  sufficient.  A  blank 
may  be  so  long  or  so  wide  as  to  be  hard  to  understand  at  one  look. 

b.  Use  "  double  distribution  "  tables  where  possible. 

By  these  are  meant "  two-way "  tables,  —  tables  that  classify  data 
two  ways,  across  the  page  on  one  classification,  and  down  the  page 
on  another.  They  thus  enable  one  to  get  the  data  from  at  least  two 
separate  tables  condensed  into  a  convenient  form  in  one.  Tables  7, 
8,  and  9,  with  slight  modifications,  are  taken  from  Snedden  and 
Allen's  School  Reports  and  School  Efficiency  1  to  illustrate  double  dis- 
tribution tables. 

Table  7.     Blank  for  Showing  Number  of  Pupils  Making 
Given  Number  of  Days'  Attendance 


First  grade  . 
Second  grade 
Third  grade 
Fourth  grade 
Etc.     .     .     . 


From       From       From       From 

20  to        !>()  to        CO  to    ,    80  to 

39  days  59  days'1  79  days\99  days 


Explanation:  This  table  provides  for  a  distribution  of  pupils' 
attendance  in  two  ways:  (1)  by  the  grade;  (2)  by  the  number  of 
days.  Hence  it  is  called  a  "two-way"  or  "double  distribution" 
table. 

1  Pp.  130,  132 


Collection  of  Data 


73 


Table  8.     Blank  for  Showing  Distribution  of  Pupils  in 
Each  Grade  by  Ages,  Records  of  Ages  Being  Made 


First  grade 

Second  grade 

Totals 

Males 

Females 

Males 

Females 

Males 

Females 

From   5  yr. 
6  mo.  to  6 

yr.  6  mo. 
From  6  yr. 
6  mo.  to  7 

yr.  6  mo. 
Etc.    .     .     . 

Note:  This  is  a  very  convenient  table,  since  it  gives  a  distribution 
of  pupils  both  by  ages,  and  by  grade  and  sex. 

c.  Strive  to  have  all  the  printing  on  the  blank  so  that  it 
can  be  read  from  one  position  while  filling  in  the  data. 

Table  7  was  originally  printed  like  Table  9. 


Table  9.     Blank  for  Showing  Number  of  Pupils  Making 
Given  Number  of  Days'  Attendance 


W 

w 

a> 

s> 

Si 

a-. 

Si 

e 

es 

3 

e 

"O 

ts 

"3 

-© 

OS 

OS 

Cs 

CS 

c^ 

G-3 

'© 

^ 

OS 

1-H 

© 

o 

o 

© 

© 

© 

© 

© 

-In 

^ 

-* 

vO 

CO 

§ 

~ 

§ 

g 

g 

© 

© 

© 

© 

o 

^ 

•— 

fe, 

fe, 

^ 

b. 

tt, 

fca 

First  grade    . 

Second  grade 

Etc 

74 


School  Statistics  and  Publicity 


Note:  This  is  exactly  the  way  the  blank  should  not  be  made  up. 
The  proper  form  is  given  on  page  72.  With  care,  the  space  usually 
can  all  be  utilized  either  way.  If  all  the  titles  and  headings  can  be 
read  from  one  position,  that  is,  without  having  to  turn  the  paper 
around,  the  chances  for  error  are  much  reduced. 

d.  Put  full  or  plainly  abbreviated  title,  not  merely  an 
arbitrary  symbol  or  a  number,  above  each  column,  if  any 
one  besides  the  superintendent  is  to  use  the  blank. 


4.  NATIONAL  SCHOOL  RECORD  SYSTEM 

TRANSFER    CARD 


To  be  filled  out  for  the  Attendance  Offi- 
cer in  case  of  transfer  to  any  other 
school,  either  in  or  outside  city.  .    .    . 


Last  name 


First  name  and  initial 


6.     Name  of  parents  or 
guardian 


8.     Residence  before 
discharge 


7.    Occupation  of  parent  or 
guardian 


9.     Date  of  discharge 


8.     New  residence  (or  name    10.    Age  when  discharged 
of  private  or  parochial  school   i 

if  pupil  is  transferred  to  one)  Years  Months 


Grade      Room 


f.  g-  h. 

Days  Present    Health    Conduct 


Scholarship 


Date  of  last  attendance 


School 


District 


Teacher 


Principal 


Remarks  on  other  side 


THE  SHAW-WALKER  CO.,    MUSKEGON       FORM    NO   4 


Some  superintendents  get  out  blanks  of  the  kind  condemned  here. 
But  it  is  very  doubtful  if  such  blanks  will  ever  secure  accuracy  from 
any  one  except  the  person'  who  thought  out  the  key  (even  he  may 
forget  it  if  it  is  at  all  intricate)  and  the  small  percentage  of  teachers 
who  may  be  called  "hyper-patient."  Teachers  dread  such  blanks 
for  precisely  the  same  reason  that  most  readers  dread  having  to  refer 
constantly  to  footnotes  in  something  they  are  trying  to  read. 


Collection  of  Data  75 

if  the  data  are  to  be  copied  or  read  to  some  one  else,  a  key  num- 
ber should  appear  in  each  column  immediately  under,  or  in  front  of, 
the  title.  The  reader  or  copyist  can  catch  this  key  number  much 
more  quickly  and  easily  than  he  can  the  heading.  The  key  number 
is  also  preferable  for  calling  or  reading  off  to  some  one  else. 

A  good  example  of  the  use  of  key  numbers  and  letters  with  titles 
is  found  in  the  National  System  of  Uniform  Records,  the  transfer 
card  of  which  is  reproduced  on  page  74. 

e.  If  the  same  item  appears  in  different  blanks  of  a 
series,  always  give  it  the  same  key  number  and  the  same 
relative  position  to  closely  connected  items. 

The  report  of  the  Committee  on  Uniform  Records  and  Reports, 
previously  referred  to,  recognized  this  principle.  The  order  of  the 
various  items  appearing  on  the  pupil's  report  card  is  the  same  as 
that  appearing  on  the  lower  half  of  the  attendance  and  scholar- 
ship record  kept  in  the  teacher's  loose-leaf  register,  which  is  later 
filed  in  the  principal's  office.  Other  cards  in  the  system  which 
contain  these  items  use  the  same  key  letters  and  relative  position. 

/.  If  data  are  to  be  summarized  from  figures  furnished  by 
many  persons,  make  the  summary  with  as  little  copying  as 
possible. 

Copying  always  tends  to  error,  and  the  checking  necessary  to  avoid 
such  errors  is  very  laborious  and  time-consuming.  The  special  econo- 
mies here  are  things  like  the  following:  (1)  In  a  building,  the  data 
for  a  report  on  the  table  given  on  page  72  may  be  secured  by  hav- 
ing each  teacher  call  at  the  principal's  office  and  put  her  data  on 
the  proper  horizontal  line.  The  completed  blank  will  then  be  the 
desired  summary.  (2)  If  the  data  are  to  come  from  different  build- 
ings, each  teacher  can  put  hers  on  the  proper  line  on  a  blank  and 
send  it  to  headquarters.  There  such  blanks  can  be  gathered,  the 
line  clipped  off  each  blank  (keeping  necessary  identification  marks  of 
course),  and  pasted  on  a  sheet  of  paper,  thus  forming  the  summary 
at  once.  In  very  extensive  or  frequent  tabulations  of  this  sort, 
blanks  with  perforations  whore  the  lines  are  to  be  cut  apart  may  be  of 
service.     Perforated  blanks  are.  however,  very  expensive  and  usually 


76 


School  Statistics  and  Publicity 


unnecessary.    For  most  purposes,  cutting  and  pasting,  or  copying 
and  checking,  will  be  sufficient. 

g.  Figure  or  summarize  on  the  blank  itself  if  time  can  be 
saved  thereby,  as  is  often  the  case. 


3abJeot         rhwS<*CS 


Teache 


r      U 


Tears  Experience,   before  1913-14_ 
Head  of  Department? 


Laea      *-} 


3X        M 


: 95-100    :    90-94    : 


65-79    :    150-701    :    76A    I    Below   65    :    Total 


Boye   : 

/ 

I 

< 

1  / 

;      1 

i     ° 

/ 

I3L 

^■*" 1 

/ 

© 

/ 

© 

i ' 

6 

57 

Girle;       / 

/// 

// 

//// 

!  / 
j© 

0 

3 

O 

0 

to 

i  ® 

1 

7 
@) 

/ 

® 

1 

Lita  for  Grading  System  -  Hume-Fogg  High  School 
Grades  for  1913-14. 

FIO,   1.  —  Example  of  Scoring  and  Figuring  on  the  Blank  Used  for  Collect- 
ing Data. 


Collection  of  Data  77 

In  summarizing  the  material  collected  as  advised  in  section  /,  the 
additions  can  be  made  in  ink  (often  red  ink)  on  the  bottom  of  the 
sheet  where  all  the  data  have  been  collected.  A  good  illustration 
of  this  point  is  afforded  by  a  blank  which  was  used  by  a  class  of 
graduate  students  making  under  the  writer's  direction  a  study  of 
the  marks  given  in  the  Hume-Fogg  High  School  in  Nashville.1  (See 
Figure  1.) 

In  Figure  1  original  scores  are  indicated  by  the  marks  above  the 
line.  The  plain  figures  below  each  line  are  the  numerical  equiva- 
lents of  the  scores.  The  figures  in  rings  represent  the  approximate 
percentage  values  of  the  numbers  just  above  them.  On  the  original 
blank  such  percentages  could  be  indicated  by  red  ink  without  using 
rings.  All  percentages  are  calculated  here  to  show  the  method.  But 
in  this  particular  problem,  because  of  the  small  number  of  cases  in 
each  group,  such  percentages  have  little  value.     See  pages  13,  63. 

h.  In  case  only  one  copy  of  a  blank  can  be  made,  use 
cross-section  paper,  or  ruled  blank  book. 

Cross-section  paper  saves  an  enormous  amount  of  time  in  ruling 
paper  and  avoids  eyestrain.  Such  paper  may  be  purchased  from  a 
good  scientific  supply  or  drawing  company,  in  sheets  of  various  sizes. 
That  particular  kind  characterized  by  heavy  lines  every  five  or  ten 
small  squares  is  the  better  kind.  The  heavy  lines  serve  as  columns, 
while  the  small  squares  help  to  keep  figures  under  each  other. 

A  ruled  blank  book  or  account  book,  especially  the  kind  extending 
over  two  adjacent  pages,  is  similarly  useful  because  of  the  vertical 
columns  and  horizontal  lines. 

i.  Put  at  the  top  of  the  blank  the  item  which  determines 
the  place  of  the  blank  in  a  series  or  which  identifies  it. 

The  observance  of  this  caution  will  make  for  ease  in  handling  the 
data.  For  example,  in  the  case  of  cards  that  are  to  be  filed  alphabet- 
ically according  to  the  name  of  the  student,  this  name  should  appear 
at  the  top  of  the  card  where  it  will  be  the  first  thing  the  eye  of  the 
one  going  through  the  records  will  light  upon.     Yet  many  school  men 

1  The  results  of  this  investigation  appeared  in  Educational  Ad- 
ministration and  Supervision,  1 :  648  (Dec,  1915),  from  which  article 
this  illustration  is  adapted. 
t 


78  School  Statistics  and  Publicity 

get  out  blanks  with  the  name  of  the  school,  the  name  of  the  card,  the 
name  of  the  town,  and  other  insignificant  items  at  the  top,  which  is 
the  choice  position  for  identifying  a  blank  quickly,  especially  if  it  is 
ever  filed.  Such  material  is  very  seldom  handled  except  by  persons 
who  use  the  blanks  more  or  less  constantly.  For  these,  this  material 
may  as  well  come  at  the  bottom  of  the  card  and  often  even  in  fine 
print.  Even  the  transfer  card  of  the  National  School  Record  System 
shown  on  page  74  could  be  improved  upon  in  this  respect.  The 
card  is  to  be  filed  alphabetically  by  the  name  of  the  pupil.  This 
name  should  come  on  the  top  line,  and  the  title  of  the  card  with  the 
caution  to  the  attendance  officer  should  be  placed  at  the  bottom. 

j.  Always  use  blanks  cut  to  standard  sizes. 

This  is  essential  for  easy  handling  of  data,  filing  for  ready  refer- 
ence, and  storage  for  future  use.  Folders  and  filing  cases  can  easily 
be  procured  for  standard  sizes,  and  the  printer  or  supply  house  can 
furnish  paper  and  cards  for  these  sizes  at  much  lower  prices  than  for 
odd  sizes.     The  standard  sizes  are  3x5,  4x6,  5x8,  and  8^x11  inches. 

Examples  of  Good  Blanks.  The  busy  superintendent 
may  often  save  much  time  by  getting  good  blanks  used 
in  other  systems  and  quickly  modifying  them  for  his  own 
purposes.     Such  blanks  may  be  found  in : 

1.  Snedden  and  Allen :  School  Reports  and  School  Efficiency. 
(Macmillan) 

This  discusses  the  good  and  bad  points  in  actual  blanks  from 
city  school  systems,  on  such  topics  as  the  school  plant,  expend- 
itures, the  census,  attendance,  attendance  and  ages,  promotions, 
summaries,  etc. 

2.  Report  of  the  National  Education  Association  Committee  on  Uni- 
form Records  and  Reports,  published  as  Bulletin  U.  S.  Bureau  Educa- 
tion, 1912,  No.  .'5. 

It  gives  model  blanks  for  the  following  in  elementary  schools : 

1.  Principal's  term  report :  (a)  enrollment ;  (o)  promotions, 
non-promotions,   by  grades;     (c)   distribution  of  withdrawals 


Collection  of  Data  79 

by  ages  and  causes ;  (d)  distribution  of  attendance ;  (e)  grad- 
uates by  years  in  schools;  (/)  non-promotions  by  grades  and 
causes ;  (g)  failures  by  studies  and  grades ;  (h)  distribution  of 
leavings  and  withdrawals  by  ages  and  grades ;  (i)  ages  of  grad- 
uates; (j)  enrollment  and  attendance;  (k)  distribution  of 
whole-time  teachers. 

2.  Teacher's  term  report:  (a)  enrollment  by  divisions; 
(6)  non-promotions  by  grades  and  classes ;  (c)  failures  by  grades 
and  studies ;  (d)  enrollment  and  attendance ;  (e)  distribution 
of  enrollment  by  ages ;  (/)  distribution  of  withdrawals  by  ages 
and  causes ;  {g)  distribution  of  leavings  by  ages  ;  (h)  beginners 
by  training;    (i)  beginners  by  ages. 

There  is  also  a  set  of  blanks  for  high  schools  similar  to  these. 

3.  Strayer  and  Thorndike :  Educational  Administration.  (Mac- 
millan) 

This  contains  extracts  from  statistical  researches  on  educa- 
tion made  at  Teachers  College,  Columbia  University.  It  con- 
tains some  of  the  blanks  recommended  by  the  National  Educa- 
tion Association  Committee  as  well  as  very  valuable  table 
forms  on  other  things. 

4.  Rugg,  H.  0. :  Statistical  Methods  Applied  to  Education,  pages 
28-87.     (Houghton  Mifflin) 

5.  Certain  Surveys. 

The  Butte  Survey,  for  example,  contains  good  suggestive 
blanks  on  these  phases :  census,  attendance,  absences,  educa- 
tion and  experience  of  teachers,  enrollment,  promotion  and 
failures,  size  of  classes,  ages  of  children  at  beginning  of  first 
semester,  receipts  and  expenditures. 

One  Blank  versus  Card  Index.  In  many  pieces  of 
statistical  work,  one  must  early  face  the  question  as  to 
whether  it  is  better  to  put  all  the  data  on  one  large 
blank,  allowing  one  line  to  each  case  or  group,  or  to  enter 
each  case  or  group  on  a  separate  card. 

One  of  the  best  arguments  for  the  latter  method  is  that 
the  data  in  the  columns  cannot  show  their  full  meaning, 


80  School  Statistics  and  Publicity 

as  a  rule,  unless  arranged  in  order  from  lowest  to  highest. 
If  the  data  are  all  on  one  large  sheet,  the  classes  may  come 
in  order  for  the  data  in  the  first  column,  but  this  order- 
will  not  be  the  same  in  any  one  of  the  other  columns, 
which  may  be  several  in  number.  If  these  columns  are 
to  be  accurately  studied,  the  data  in  them  must  be 
copied  off  and  rearranged,  with  many  chances  for  error. 
On  the  other  hand,  if  no  large  table  is  made,  the  cards 
may  be  taken  up  in  order  easily  for  one  column.  The 
data  may  then  be  copied  off  for  a  table  on  this  one  item. 
Next,  the  cards  may  be  rearranged  for  column  two  and 
the  corresponding  table  made.  This  process  is  much 
easier  mentally  and  more  accurate  than  the  first  one 
mentioned  above.  It  is  the  procedure  used  in  the  Courtis 
tests,  where  each  child  has  a  folder  containing  his  own 
work.  These  folders  are  taken  up  and  sorted  into  piles 
of  differing  achievement  for  attempts  in  addition,  and 
from  these  a  distribution  table  is  made ;  they  are  re- 
distributed for  attempts  in  subtraction,  and  a  second 
distribution  table  is  made,  this  time  for  subtraction ;  the 
process  is  repeated  for  multiplication  and  division.  Then 
the  start  is  made  on  "rights,"  and  the  four  sortings  take 
place  on  this  basis  and  the  four  resulting  tables  are 
copied  off. 

To  summarize,  the  advantages  of  each  plan  are  : 

Large  Sheet  Card  Index  Plan 
The  data  are  not  easily  lost.               Less  chance  for  error  in  copy- 
It  is  impossible  to  lose  a  part,  ing  or  rearranging  data. 

as  can  easily  happen  in  the  use  Easier  mentally. 

of  cards. 

All  data  are  before  the  eye  at  Any  parts  of  the  data  may  be 

one  time,  and  thus  a  better  pre-  separated  from  the  remainder  at 

liminary  grasp  of  the  situation  any  time. 

may  be  had. 


Collection  of  Data 


81 


Less    work    in    compiling    at  The  data  may  be  easily  shifted 

first.  to  any  order. 

At  the  beginning  data  may  be  Pupils  can  handle  data  better, 

copied  in  any  order.  Often  it  will  be  advisable  to  give 

portions  to  different  workers. 

The  disadvantages  of  these  two  plans  are  largely  the 
converse  of  the  advantages,  but  they  may  be  emphasized 
through  the  statement  of  them : 


Card  Index  Plan 
If  card  is  lost,  it  cannot  be  re- 
placed, nor  can  the  loss  be  easily 
discovered. 

A  card  may  stick  to  another 
and  be  easily  overlooked. 

Cards  for  easy  handling  must 
ordinarily  be  kept  in  alphabetical 
order.  Hence  if  shifted  for  pur- 
poses of  making  tables,  etc., 
must  be  returned  to  regular  order. 

Much  mechanical  work  is  neces- 
sary in  examining  data. 


Large  Sheet 

Data  from  large  buildings  or 
groups  not  easily  brought  to- 
gether on  one  sheet. 

Additional  data  (often  come 
late)  are  hard  to  insert. 

Data  in  all  columns  not  ar- 
ranged from  highest  to  lowest 
nor  can  they  be  so  arranged  with- 
out recopying. 

More  chances  for  error  in  re- 
copying  and  rearranging. 

Will  stand  less  wear  and  tear. 

Averages  and  summaries  are 
hard  to  make  unless  data  are  ar- 
ranged in  order  from  the  start. 
Often  this  cannot  be  determined 
when  data  are  copied  on  large 
sheet. 

On  any  given  problem,  a  little  forethought  or  practice 
on  sample  data  will  usually  indicate  which  is  the  better 
plan.  However,  it  is  very  difficult  to  be  certain  in  this 
matter  until  the  whole '  statistical  procedure  has  been 
thought  through. 

EXERCISE 

For  your  special  problem  : 
(a)  Decide  just  where  you  would  use  a  card  index  and  just  where 
one  large  blank,  giving  full  reasons. 


82 


School  Statistics  and  Publicity 


(b)  List  the  blanks,  with  very  definite  titles,  that  you  now  think 

you  would  need. 

(c)  Draw  off  in  complete  detail,  accurate  to  size  of  paper,  at 

least  one  of  these  blanks. 

5.    Miscellaneous  Economies  in  Collecting  Data 

a.  Cross-section  paper  for  scoring.  Many  times  it  is  necessary  to 
keep  track  of  the  number  of  cases,  the  data  for  which  may  come  from 
widely  separated  places.  Keeping  track  mentally  is  very  difficult 
and  inaccurate.  It  is  far  better  to  adopt  some  method  of  putting 
down  a  mark  for  each  case  as  soon  as  it  is  located,  and  later  count 
the  marks.     The  cross-section  paper  device  is  one  for  such  marking. 


1 

1 

1 

1 

1 

1 

1 

1 

1 

c 

1 

1 

1 

1 

1 

1 

1 

*0 

-1 

oc 

) 

1 

1 

1 

1 

1 

1 

1 

1 

1 

I 

1 

J 

1 

50 

~ 

81 

5 

., 

Fig.   2.  —  Scoring  on  Cross-Section  ISpcr. 

Suppose  that  a  study  is  being  made  of  the  marks  given  by  various 
teachers  to  pupils.  A  sheet,  column,  or  whatever  space  is  desired, 
may  be  assigned  to  each  teacher.  One  horizontal  row  of  large  squares 
may  he  labeled  90  -100  or  A,  according  to  the  grading  system  ;  another 
80  89  or  R,  etc.  Each  mark  between  90  and  100  as  soon  as  located 
would  Ik?  represented  by  a  check  mark  in  one  of  the  little  squares  in 


Collection  of  Data  83 

the  row  of  large  squares  representing  grades  90-100,  under  the  column 
for  the  proper  teacher  or  subject.  For  such  work,  always  begin  at 
the  left  upper  corner  of  the  first  big  square  on  the  left  and  fill  in  reg- 
ularly to  the  right,  row  by  row.  Suppose,  when  the  work  has  been 
finished,  that  the  90-100  column  looks  like  the  illustration  given. 
One  glance  at  the  squares  in  Figure  2  shows  that  there  are  33  grades 
between  90  and  100  given  by  the  teacher  during  the  time  under  in- 
vestigation. 

b.   Scoring  by  fives  on  plain  paper.     This  is  the  old  device  used  in 
counting  the  votes  in  elections,  etc.     The  first  four  cases  of  every  five 

m 

m 

THl 

m  or  w  m  m  m  m    mi  m 
m 


Tttt. 

in 


Fig.   3.  —  Scoring  on  Plain  Paper. 


are  given  a  straight  perpendicular  mark  each ;  the  fifth  is  made  hori- 
zontally, tying  the  other  four,  as  in  Figure  3. 

This  makes  33  cases,  as  in  the  preceding  method  of  scoring.  For 
most  people  the  counting  is  much  easier  if  the  groups  of  five  are  under 
each  other  and  not  in  a  horizontal  line. 

The  author  also  has  made  much  use  of  the  scheme  of  putting  down 
a  m;  rk  for  every  case  in  the  group  as  best  he  could  for  speed,  on  any 


Fig.  4.  —  Scoring  on  Plain  Paper.' 

paper  handy.     Then  rings  are  drawn  around  the  marks  with  five  in 
each  ring-     It  is  easy  to  count  the  rings.     The  chief  merit  of  this 


84 


School  Statistics  and  Publicity 


scheme  for  a  single  worker  is  that  the  eye  does  not  have  to  be  shifted 
quickly  and  accurately,  with  the  resulting  strain.  Putting  down  the 
dots  roughly  requires  no  eyestrain.  Putting  the  rings  around  the 
dots  is  not  hard,  as  the  eye  is  kept  focused  on  the  particular  part  of 
the  paper.  Thus,  the  same  33  cases  might  be  scored  as  in  Figure  4. 
c.  Checking  on  blanks.  Where  possible,  tabulate  by  check  marks 
in  appropriate  column  on  a  blank,  as  in  the  United  States  Govern- 
ment stock  device..  Thus,  a  superintendent  could  study  physical 
defects  of  school  children  with  a  blank  like  Table  10. 

Table    10.    Blank   Illustrating    Method    of    Checking    in 
Entering  Data 

Physical  Defects 


Pupil  No. 

Sight                 Hearing 

Ade- 
noids 

Tubercu- 
losis 

Etc. 

Rigid 

Left 

Right 

Left 

1 

2 
3 
4 
5 
6 
7 

X 

X 
X 

X 
X 

X 

X 

d.  Some  entry  for  each  case.  In  some  instances  accuracy  is  in- 
creased by  the  device  of  making  some  sort  of  entry  for  each  case.  It 
is  very  easy  to  omit  a  few  cases  from  a  large  number  if  entries  are 
made  only  where  data  actually  exist.  But  if  numbers  are  put  down 
for  actual  data,  zeros  if  it  is  known  that  nothing  is  done,  and  "n.  d." 
'for  "no  data")  where  it.  has  been  impossible  to  secure  data,  the 
results  will  be  likely  to  be  much  more  accurate.  The  necessity  of 
having  to  account  for  each  case  in  a  positive  form,  reduces  the  chances 
for  omission,  a  fact  long  ago  discovered  by  insurance  companies, 
which  require  their  agents  to  make  some  report  on  each  item  on  a 
blank. 


Collection  of  Data 


85 


e.  Ruler  strip  for  printed  reports.  Often  it  is  necessary  for  the 
superintendent  to  get  figures  from  two  or  three  widely  separated 
columns  in  a  table  of  fine  print.  There  is  great  eyestrain  and  many 
chances  for  inaccuracy  in  trying  to  copy  them  off  directly,  especially 
where  the  tables  run  over  two  pages,  as  do  those  in  the  government 
bulletins.  The  best  way  to  avoid  this  is  to  cut  out  a  strip  of  paper 
which  may  be  placed  over  the  table  so  that  the  desired  figures  will 
stand  out  clearly  and  quickly  in  the  angles  of  the  paper.  Table  11 
indicates  how  to  work  the  scheme  if  the  superintendent  wishes  to 
compare  his  system  with  the  cities  in  the  table  in  the  matter  of  ex- 
penditures under  the  headings,  Board  of  Education  and  Business 
Offices,  Superintendent's  Office,  and  Other  Supplies.  Government 
offices  use  specially  constructed  rulers  for  the  same  purpose. 

Table  11.     Illustration   op  Use   of   Ruler   Strip  Device  on 

a  Page,  in  the  Report  of  the  U.  S.  Commissioner  of 

Education 


Cities 

B'd  of  Ed. 
&Bus. 
Offices. 

Svjpf  5 

Office. 

Sal 
<Sl  Exp 

of 
Super- 
visors 

Sal 

&£xp. 

of 

Prm. 

Sal.of 
Teachers 

Text 
Books 

Other 
Supplies 

ALABAMA 

$  3631 
128163 

$10750 
43801 

$5967 
IO5600 

Birmingham 
CALIFORNIA 
^osAngeies 

1 

Another  very  convenient  form  of  ruler  strip  is  made  with  two 
strips  of  zinc  adhesive  or  bicycle  tape  and  some  thin,  tough  paper. 
The  strips  of  adhesive  are  pasted  across,  the  paper,  but  a  little  apart. 
Then  the  top  strip  is  cut  across,  the  cuts  corresponding  to  the  lines  in 
the  table  to  be  used.  Then  some  of  the  resulting  flaps  may  he  pulled 
back  and  pressed  back  to  show  the  figures  in  the  desired  columns, 
as  in  Figure  5. 


86 


School  Statistics  and  Publicity 


The  advantages  of  this  form  are  that  it  can  be  used  for  various 
combinations  of  columns,  and  that  it  is  very  durable. 

/.  One  operation  at  a  time.  The  idea  is  to  carry  the  same  step 
or  operation  all  the  way  through  without  stopping  to  do  something 
else.  That  is,  if  one  is  preparing  a  table  from  data  found  in  different 
tables,  he  should  copy  all  the  data  from  each  table  in  turn,  and  not 
skip  from  one  to  the  other.     Or  if  one  has  several  groups  of  data  to 


?a 


per^ 


Jape 


43801 

1               1 

1               ' 

1               1 
■               i 

105600 

i 
1 
1 

—     — 

—  —  •— 

Fig.  5.  —  Illustration  of  Use  of  Adhesive  Tape  Ruler  Strip  on  Figures  of 

Table  11. 


rearrange,  he  should  finish  each  group  before  going  to  the  next.  This 
gives  the  practice  effect  for  that  operation,  insures  greater  accuracy, 
and  is  much  easier  mentally. 

g.  Using  high  school  students  to  gather  data.  For  gathering  many 
of  the  data,  high  school  students  can  do  as  well  as  any  one  else.  A 
class  can  collect,  classify,  and  check  a  great  deal.  Of  course  the 
question  at  once  arises  as  to  whether  this  is  a  legitimate  use  of  the  time 
of  high  school  students,  who  supposedly  go  to  school  for  their  own 
benefit  and  not  to  help  work  out  school  statistics.  But  many  of  these 
are  later  to  do  clerical  work  of  various  sorts.  Much  of  the  statistical 
work  in  school  affords  the  finest  sort  of  clerical  practice  for  such  stu- 
dents. Their  welfare  will  be  properly  cared  for  if  the  work  is  done 
under  the  careful  supervision  of  the  superintendent.  He  will  natu- 
rally discover  a  great  deal  of  valuable  knowledge  about  the  vocational 
aptitudes  of  his  individual  students  for  satisfactorily  doing  clerical 
work  or  for  directing  others  in  such  work.  Accordingly  it  may  be 
considered  thoroughly  sound  preliminary  training  for  them,  as  Pro- 
fessor Bobbitt  has  pointed  out.1 

But  there  is  more  direct  evidence  of  the  value  of  such  work  than 
any  such  theoretical  statement.  In  1915  one  of  the  author's  graduate 
students,  Mr.  S.  J.  Phelps,  later  professor  of  secondary  education  at 
the  University  of  Vermont,  directed  the  work  of  the  students  of  the 

1  San  Antonio  Survey,  pp.  32,  33, 


Collection  of  Data  87 

Gallatin,  Tennessee,  High  School  in  making  a  study  of  four  problems 
connected  with  their  school  system.1     These  problems  were : 

1.  Cost  of  maintenance  of  the  system. 

2.  Age-grade  distribution  of  the  pupils. 

3.  Variations  in  marks  given  by  high  school  teachers. 

4.  Study  of  the  lighting  facilities  in  each  room. 

Clear  and  concise  instructions  were  placed  by  Professor  Phelps  in  the 
hands  of  the  various  teachers  engaged  in  the  work.  The  answers  of 
the  students  were  carefully  checked  over  in  class  and  then  submitted 
to  the  inspection  of  a  graduate  class  in  school  administration  at  George 
Peabody  College  for  Teachers.  Most  of  these  men  had  had  much 
experience  in  administrative  work  and  all  were  specializing  in  survey 
work.  They  unanimously  agreed  that  the  work  was  surprisingly 
accurate. 

In  answer  to  the  question  :  Is  not  such  work  an  exploitation  of  the 
students  ?  Professor  Phelps  answers : 

"All  are  agreed  that  in  mathematics,  especially,  it  is  necessary 
for  a  high  school  student  to  do  a  large  amount  of  drill  work.  Now 
in  doing  this,  which  is  the  more  profitable  and  practical  for  a  high 
school  student  who  is  studying,  for  instance,  eleventh  grade  civics 
or  arithmetic,  to  do  as  outside  work :  To  study  the  costs  of  oper- 
ating his  own  school  system,  compared  to  similar  costs  in  other 
towns,  or  to  find  out  the  number  of  days  it  will  take  A,  B,  and  C 
working  together  to  do  a  piece  of  work  which  A  can  do  in  three 
days,  B  in  four  days,  and  C  in  five  days? 

"How  would  a  study  of  costs  in  his  school  compare  with  a  paper 
which  he  might,  after  much  delving,  prepare  on  the  source  of  some 
abstract  principle  of  governmental  costs?  Isn't  this  a  place  where 
the  much-talked-of  subject,  Community  Civics,  could  get  some 
practical  problems? 

"Which  seems  more  practical  and  profitable  for  a  class  in  algebra 
studying  the  graph,  —  to  make  a  graph  of  these  same  costs,  or  to 
graph  the  profile  of  a  river  bed,  or  perhaps  an  extract  from  the 
table  of  American  Mortality  Experience? 

"In  which  would  a  student  in  percentage  be  expected  to  show 
the  more  interest,  —  in  a  study  of  the  percentage  distribution  of 

1  Phelps,  S.  J. :    Master's  thesis  at  George  Peabody  College  for 
Teacher^,  1915,  on  file  in  library. 


88  School  Statistics  and  Publicity 

the  marks  given  by  his  own  teachers,  among  which  he  has  a  mark, 
or  in  studying  the  percentage  composition  of  some  compound, 
perhaps  a  fertilizer,  which  he  has  never  seen  and  in  which  he  cannot 
be  expected  to  show  a  passing  interest  or  curiosity? 

"Would  another  student  in  practical  measurements  get  more 
from  computing  the  surface  and  volume  of  the  earth,  or  from  find- 
ing how  many  spheres  of  a  certain  diameter  could  be  placed  in  a 
cylindrical  cup  of  certain  dimensions,  than  he  would  get  from 
studying  the  ratio  of  lighting  space  to  floor  space  and  air  space  per 
pupil  in  the  same  room?" 

From  such  suggestive  questions  as  these  it  may  be  surmised  that 
work  of  this  sort,  instead  of  "exploiting"  the  high  school  student, 
would  be  of  great  practical  benefit  to  him. 

A  report  of  similar  work  from  Wisconsin  is  as  follows : 

HIGH    SCHOOL    CLASSES 
GRAPH    CONDITIONS 

Over-Age  and  Failures  Studied 

In  connection  with  the  algebra  work 
in  the  Frederic  High  School,  graphic  rep- 
resentations in  colors  are  made  showing 
conditions  in  the  number  of  students  re- 
tarded, and  other  school  problems.  One 
percentage  graph  compares  the  percentage 
of  students  over  age,  showing  a  distinct 
decrease  in  retardation  during  the  past 
four  years.  Other  graphs  include  mate- 
rial on  students  dropped,  failed,  and  pro- 
moted in  various  subjects.  Teachers  are 
also  compared  with  respect  to  the  number 
of  students  failed  by  each  of  them. 

Incidentally,  such  work  as  this  gives  the 
algebra  pupil  practical  work  to  do,  illumi- 
nates the  general  subject  (if  graphic  analy- 
sis, and  makes  mathematics  interesting.1 

'Wisconsin  State  Department  of  Education:  Educational  News 
Ihilirth,,  Jan.  1,  1  <J  1 7 . 


Collection  of  Data  89 

EXERCISE 

Which  of  the  miscellaneous  economies  would  be  applicable  to  your 
problem,  and  just  how  would  you  use  them? 

REFERENCES    FOR   SUPPLEMENTARY    READING 

King,  W.  I.     Elements  of  Statistical  Method,  Chapters  IV-IX. 
Report  of  the  Committee   on    Uniform  Records  and  Reports.      U.  S. 

Bureau  of  Education  Bulletin,  1912,  No.  3. 
Rugg,  H.  O.     Statistical  Methods  Applied  to  Education,  Chapters  II, 

III. 
Thorndike,  E.  L.     Mental  and  Social  Measurements,  Chapter  II. 


CHAPTER   III 

TECHNICAL   METHODS   NEEDED   IN   SCHOOL 
STATISTICS 

I.    USUAL  VIEWS 

So  far  we  have  considered  only  statistical  matters 
that  are  plain  to  any  experienced  school  man.  With  the 
suggestions  previously  given,  such  a  man  could  success- 
fully collect  statistical  data  on  most  of  his  school  prob- 
lems. But  the  working  up  of  the  data  and  the  proper 
interpretation  of  them  would  be  altogether  different 
and  much  more  difficult.  He  would  at  once  face 
such  questions  as  these :  Does  the  superintendent  need 
any  technical  knowledge  of  statistics?  Can  he,  without 
such  special  knowledge,  analyze  his  data  and  get  the 
really  significant  things  out  of  them?  Or,  without  such 
knowledge,  can  he  present  these  results  effectively  to  the 
public  ? 

The  Conservative's  View.  Attempts  to  answer  these 
questions  have  brought  forth  much  nonsense  and  fruit- 
less effort.  For  example,  one  group  of  school  men  take 
the  position  that  a  superintendent  needs  iTo  special  knowl- 
edge of  statistics,  simply  because  there  is  in  their  opinion 
no  virtue  in  statistics.  They  quote  the  old  statement  of 
Bagehot :  '  There  are  three  kinds  of  lies  —  lies,  damned 
lies,  and  statistics."  Or  they  reiterate :  "  Figures  don't 
lie,  but  liars  do  figure."     Or  else  they  would  at  least 

90 


Technical  Methods  in  School  Statistics     91 

agree  with  the  author  of  a  recent  article  that  "  whatever 
the  causes,  the  fact  is  that  any  one  who  presents  his 
arguments  in  the  form  of  tables,  and  his  conclusions  in 
dogmatic  statements  presumably  based  on  the  tables, 
is  sure  to  convince  nine  tenths  of  his  readers."  1  Con- 
sequently, superintendents  having  this  viewpoint  see  no 
need  of  any  special  knowledge  of  statistics.  They  think 
that  any  effort  to  do  anything  unusual  with  statistical 
data  is  simply  a  waste  of  time.  The  absurdity  of  such  an 
opinion  should  be  evident  to  any  one  who  has  even  glanced 
through  the  preceding  chapters  of  this  book.  Professor 
King  states  the  whole  point  very  strikingly  thus :  "To 
attempt  to  handle  statistics  properly  without  a  knowledge 
of  statistical  method  is  only  a  little  less  absurd,  though 
vastly  more  common,  than  to  attempt  to  build  a  great 
steel  bridge  without  a  knowledge  of  trigonometry."  2 

The  Specialist's  View.  Another  group,  composed  of 
scientific  educators  and  statistical  experts,  advocate 
either  a  very  thorough  course  in  statistical  method  or 
none  whatever.  They  quote  Pope  on  "  A  little  learning 
is  a  dangerous  thing,"  etc.  Or  they  say  that  giving  a 
school  man  only  a  little,  or  very  superficial,  knowledge  of 
statistics  is  like  putting  a  razor  in  the  hands  of  a  baby. 
Or  they  compare  the  results  of  such  a  procedure  with  those 
ensuing  when  very  delicate  and  expensive  machinery  is  put 
in  the  hands  of  a  novice.  Such  machinery,  which  would 
produce  wonders  if  run  by  a  competent  man,  is,  of  course, 
soon  ruined  by  a  bungler,  and  the  result  produced  is  very 
inferior  or  altogether  lacking.  Even  Professor  King  says  : 
"  The  science  of  statistics,  then,  is  a  most  useful  servant, 

1  "Lies,  Damned  Lies  and  Statistics,"  Unpopular  Review,  1915. 
Vol.  II,  352-353 

2  King<  W.  I. :  Elements  of  Statistical  Method,  pp.  37-38 


92  School  Statistics  and  Publicity 

but  only  of  great  value  to  those  who  understand  its 
proper  use."  l 

The  Golden  Mean.  The  truth  probably  lies  between 
these  extremes.  Even  Professor  Thorndike,  the  pioneer 
in  the  application  of  modern  statistical  method  to  edu- 
cational problems,  favors  this  moderate  view  when  he 
says :  "  There  is,  happily,  nothing  in  the  great  principles 
of  modern  statistical  theory  but  refined  common  sense, 
and  little  in  technique  resulting  from  them  that  general 
intelligence  cannot  readily  master." 2  Of  course,  he 
later  says  that  mathematical  gifts  and  training  will  be 
very  useful  to  students  of  quantitative  mental  science, 
but  such  things  are  not  absolutely  necessary  for  learning 
the  elements  of  statistical  method. 

Observations  in  other  fields  also  support  this  "  golden 
mean  "  view.  Often  the  successful  politician,  minister, 
or  business  man  has  better  practical  ways  of  controlling 
people  and  reading  human  nature  than  has  the  expert  in 
the  psychology  of  such  matters.  It  is  not  an  uncommon 
thing  for  an  experienced  public  speaker  to  influence  an 
audience  more  than  a  teacher  of  public  speaking  could 
hope  to  do.  We  are  only  saying  in  other  words  that 
common  sense  and  first-hand  experience  in  reading  and 
controlling  human  minds  are  as  powerful  factors  in  in- 
fluencing the  public  as  expert  knowledge  in  the  mechanics 
of  any  art  having  the  same  end  in  view.  Is  it  not  reason- 
able, then,  to  expect  the  experience4  superintendent 
with  a  small  amount  of  statistical  theory  to  outdistance, 
in  practical  statistics  with  the  public,  the  best-trained 
experts  in  statistical  theory  only? 

1  King :   np.  cit.,  p.  33 

'*  Thorndike,  E.  L. :   Mental  and  Social  Measurements,  p.  2 


Technical  Methods  in  School  Statistics     93 

II.     STATISTICAL  KNOWLEDGE  NEEDED  FOR  SCHOOL 
SURVEYS 

But  another  phase  of  this  question  arises.  The  modern 
superintendent  must  be  able  to  read  and  understand 
school  surveys  and  apply  the  statistical  methods  used  to 
his  own  school  problems.  How  much  statistical  work 
does  he  need  for  this  sort  of  thing? 

A  study  of  practically  every  school  survey  thus  far 
published  shows  that  to  meet  this  requirement  the 
superintendent  should,  in  addition  to  things  previously 
mentioned,  know: 

1.  The  meaning  of  these  terms:  median,  average,  quartile,  range, 
central  tendency,  variability,  overlapping,  and  coefficient  of  varia- 
bility. 

2.  The  methods  of  computing  medians,  quartiles,  averages,  and 
variability  so  that  any  fallacies  or  mistakes  arising  from  poor  or  false 
methods  may  be  detected. 

3.  How  to  read  and  understand  tables  and  graphs. 

4.  The  principles  underlying  the  theory  of  good  and  bad  units,  and 
how  the  unit  in  the  particular  study  was  derived. 

5.  The  principles  underlying  the  construction  of  graphs,  so  that 
he  will  not  be  misled  by  a  badly  constructed  graph. 

6.  How  scales  are  derived,  so  that  he  will  not  be  misled  by  the 
interpretations  made  from  data  measured  on  these  scales. 

However,  the  citation  of  survey  material  as  evidence 
of  the  relatively  small  amount  of  statistical  knowledge 
needed  by  the  superintendent  is  open  to  at  least  one 
criticism,  that  the  surveys  are  written  for  the  public,  and 
to  be  effective  must  contain  little  or  no  technical  material. 
But  the  proper  translation  of  such  facts  into  popular 
language  presupposes  accurate  statistical  work.  And 
few  superintendents  could  hope  to  do  more  than  to  give 
the  best  possible  statistical  treatment  to  such  problems 
as  are  sp  treated  in  all  our  best  school  surveys  to  date. 


94  School  Statistics  and  Publicity 

III.     STATISTICAL     KNOWLEDGE     NEEDED     FOR     READING 
EDUCATIONAL   INVESTIGATIONS 

Moreover,  for  publicity  work  a  good  superintendent 
needs  to  keep  up  with  recent  investigations  in  education 
and  psychology,  as  carried  on  by  investigators  in  the 
various  schools  of  education  and  by  educational  founda- 
tions. Some  of  the  most  valuable  of  these  are  not  written 
especially  for  laymen  and  require  some  knowledge  of 
statistical  terms  and  methods  to  be  understood  at  all. 
Good  abstracts  or  news  items  of  such  studies  often  employ 
these  terms.  The  superintendent  needs  to  understand 
the  terms  and  general  processes  used,  but  he  need  not 
know  how  to  calculate  or  employ  these  terms  accurately 
himself.  It  would  be  better  if  he  could  so  use  them,  but 
it  is  not  absolutely  necessary.  The  situation  is  similar 
to  that  in  reading  for  all  of  us.  We  must  know  how  to 
employ,  use,  and  spell  correctly  certain  words.  We  need 
to  be  able  to  read  with  sufficient  understanding  a  much 
larger  list  of  words.  But  it  is  not  necessary  for  us  to 
know  how  to  spell  these  latter  words,  or  how  to  use 
them  correctly  in  our  speech  or  writing. 

The  superintendent  needs  to  understand  the  meaning, 
for  reading  purposes,  of  at  least  the  following : 

1.  Such  terms  as :  average  deviation,  standard  deviation,  mode, 
probable  error,  inter-percentile  range,  correlation,  skewness,  dis- 
persion. Xr 

2.  The  reliability  of  the  various  measures  of  variability. 

3.  The  effect  of  different  methods  of  grouping  data,  on  the  con- 
clusions reached. 

4.  Some  of  the  common  methods  of  making  allowance  for  the  un- 
reliability of  data. 


Technical  Methods  in  School  Statistics     95 

IV.     ILLUSTRATION    OF    VALUE    OF    STATISTICAL    METHOD 
TO    THE    SUPERINTENDENT 

The  value  of  statistical  method  and  presentation  to  the 
superintendent  may  be  most  clearly  presented  by  a  con- 
crete illustration.  Suppose  a  superintendent  wishes  to 
know  whether  his  high  school  classes  are  too  large  or 
too  small  for  good  work.  He  may  take  as  his  standards 
the  pronouncements  of  colleges  or  universities,  or  the 
actual  classes  found  in  high  schools  that  he  or  some 
competent  person  rates  as  good  ones. 

He  would,  of  course,  get  the  enrollments  of  all  the 
classes  in  his  own  high  school,  say  fifty  or  more.  He 
would  get  similar  figures  for  twenty  other  high  schools, 
or  more  if  possible,  say  for  at  least  a  thousand  classes. 
Then  he  would  be  confronted  with  the  problem  of  handling 
this  enormous  mass  of  data  so  as  to  bring  any  clear  idea 
out  of  it.  His  data  would  cover  pages  and  pages.  He 
would  have  an  unwieldy  mass  of  facts  that  needed  sim- 
plifying. Without  proper  treatment,  his  ideas  of  the 
whole  would  be  "  decidedly  vague  and  indefinite."  He 
would  need  some  procedure  that  would  enable  him  to 
"  give  clear-cut  form  to  this  hazy  conception  "  and  to 
"  set  objects  in  their  proper  perspectives  and  relation- 
ship." 1 

Now  if  he  knew  the  elements  of  statistical  method,  he 
could  very  shortly  summarize  his  data  on  a  half  page,  as 
Professor  Bobbitt  does  in  the  School  Review  for  October, 
1915. 2  In  this  article,  to  which  we  have  previously 
referred,  Professor  Bobbitt  is  making  studies  of  the  cost 

1  King,  W.  I. :  Elements  of  Statistical  Method,  p.  28 

2  Bobbitt,  J.  F. :   "High  School  Costs,"  School  Review,  23:    505- 

534        t 


96 


School  Statistics  and  Publicity 


of  instruction  per  one  thousand  student  hours  in  a  number 
of  high  schools.  First,  each  study,  as  English,  mathe- 
matics, etc.,  is  worked  out  separately  as  shown  in  Table 
1,  on  page  18  of  this  book.  Table  12  is  the  summarizing 
table. 


Table  12. 


Bobbitt  Table  Showing  Sizes  op  High  School 
Classes  by  Subjects 


Median 
No.  Pupils 


"Zone  of  Safety' 


Music 

Physical  Training 

English 

Mathematics   .     .     .     . 

History 

Science 

Agriculture  .  .'  .  . 
Commercial     .     .     .     . 

Drawing 

Modern  Languages  .     . 

Latin 

Household  Occupations 
Normal  Training  .  . 
Shopwork 


58 
32 
22 
21 
21 
20 
19 
19 
18 
17 
17 
17 
15 
14 


Pupils 
42-88 
28-55 
20-24 
18-24 
17-23 
16-22 
18-25 
15-23 
14-24 
15-20 
14-19 
13-23 
10-21 
12-18 


To  any  one  who  understood  the  simplest  things  about 
statistics,  this  table  would  at  a  glance  disclose  such  facts 
as  these :  In  music,  half  the  schools  ha*se  more  than  58 
pupils  in  a  class,  and  half  have  less ;  half  of  them  have 
between  42  and  88  pupils ;  a  fourth  have  less  than  42 
pupils ;  a  fourth  have  more  than  88  pupils.  Similar 
statements  would  hold  for  the  other  subjects  down  the 
table.  The  table  would  also  disclose  that  the  "  average 
classes  "    are    in    agriculture    and    commercial    subjects, 


Technical  Methods  in  School  Statistics     97 

with  19  pupils  each  as  a  rule.  Half  the  classes  in  other 
subjects  have  more  than  this  number  and  half  of  them 
have  less.  The  table  would  show  which  had  more  and 
which  less,  ranging  from  music  with  58  down  to  shopwork 
with  14.  A  little  more  inspection  would  disclose  which 
classes  ranged  more  in  their  variations  from  the  " typical' ' 
or  "  average  "  class  in  that  subject.  All  these  things,  if 
told  in  words,  would  occupy  pages  and  pages  of  description 
that  would  be  about  as  clear  and  interesting  as  a  real- 
estate  deed  to  the  lot  on  which  the  superintendent's  home 
stood. 

The  Bobbitt  table,1  of  course,  is  infinitely  clearer  and 
more  forcible  than  the  great  masses  of  data  or  the  long 
and  tedious  description  could  ever  be.  It  has  indeed 
fulfilled  "  one  of  the  prime  objects  of  statistics."  This, 
according  to  Professor  King,2  is  "  to  give  us  a  bird's-eye 
view  of  a  large  mass  of  facts,  to  simplify  this  extensive 
and  complex  array  of  isolated  instances  and  reduce  it  to  a 
form  which  will  be  comprehensible  to  the  ordinary  mind." 

V.     STATISTICAL    METHOD    AS    A    FORM    OF    EXPRESSION 

Finally,  the  superintendent  needs  statistical  method 
just  as  he  does  any  other  method  of  effective  presentation 
or  expression.  A  good  description  of  scenery,  of  an 
object,  a  face,  etc.,  always  proceeds  by  giving  first  a 
bird's-eye  view,  or  very  brief  comprehensive  sketch, 
called  usually  the  "  fundamental  image,"  or  in  exposition, 
"  the  topic  sentence."  Then  the  details  are  later  filled 
in.  The  success  of  the  description  or  explanations  depends 
upon  the  clearness,  brevity,  and  vividness  of  the  funda- 

1  The  writer  uses  this  name  because  such  tables  appear  to  have 
been  first  used  by  Professor  J.  F.  Bobbitt  of  the  University  (  '  Chicago. 

2  Kin£:    op.  tit.,  p.  22 


98  School  Statistics  and  Publicity 

mental  image,  and  then  upon  the  extent  to  which  approxi- 
mately all  the  important  details  place  themselves  clearly 
under  it.  Of  course,  the  whole  process  must  not  be  so 
mechanical  and  obvious  as  to  disgust  the  reader.  But 
without  the  fundamental  image  and  this  procedure,  the 
reader  would  soon  be  utterly  lost  in  the  details,  or  he 
would  get  so  few  of  them  in  mind,  or  in  such  a  disorganized 
manner,  that  he  would  get  no  clear  idea  of  the  whole. 
And  he  would  not  waste  any  more  time  in  trying  to  do  so. 
In  the  same  way  statistical  method  forces  a  mass  of 
numerical  data  into  a  form  which  describes  the  whole 
by  giving  a  good  fundamental  image  or  picture.  At  the 
same  time  it  leaves  all  the  data  so  grouped  and  classified 
that  significant  points  stand  out,  but  both  points  and 
minor  details  of  any  consequence  may  all  be  easily  located 
under  the  fundamental  image. 

If  suitable  graphic  presentations  of  the  statistical  results 
are  made,  they  will  have  the  same  advantage  that  a 
line  drawing  has  over  a  photograph.  The  superintendent 
without  a  knowledge  of  statistics  might  give  in  words 
only  a  picture  that  would  correspond  to  a  photograph 
taken  without  proper  focus.  This,  of  course,  gives  all 
the  details,  but  blurred.  A  suitable  graph  would  make 
the  essential  facts  of  the  whole  and  the  essential  details 
stand  out  clearly.  It  has  been  found  repeatedly  by  text- 
book writers  for  beginners  that  line  drawings  emphasizing 
the  essential  elements  in  apparatus,  pictures,  etc.,  are 
preferable  to  actual  photographs  of  the  objects,  because 
the  photographs  give  too  many  details  and  so  obscure 
the  big  things.  The  superintendent  with  no  knowledge  of 
statistical  method  could,  under  the  most  favorable  cir- 
cumstances, give  his  reader  only  a  sort  of  blurred  photo- 
graph of  his  ideas.     And,  indeed,  he  would  probably  have 


Technical  Methods  in  School  Statistics     99 

only  a  blurred  picture  of  them  in  his  own  mind.  With 
statistical  method  he  could  have  in  mind  a  sharp  line 
drawing  and  give  his  readers  the  same  kind  of  picture. 
The  essentials  of  this  method  that  are  of  value  to  him 
will  be  taken  up  in  the  next  chapter. 

EXERCISES 

1.  Which  of  the  statistical  terms  mentioned  on  page  93  do  you 
think  you  understand  fully? 

2.  Which  of  the  statistical  processes  given  on  the  same  page  do 
you  think  you  know  how  to  do? 

Note :  The  best  way  to  know  whether  you  understand  a  term  fully 
is  to  see  whether  you  can  quickly  write  a  clear  explanation  of  it. 
Similarly,  to  know  whether  you  can  tell  how  to  do  a  process,  see  if 
you  can  quickly  write  directions  for  doing  it. 

3.  Which  of  the  statistical  terms  mentioned  on  page  94  have  you 
come  across  in  your  reading,  and  just  what  do  they  mean  to  you  at 
present? 

REFERENCES   FOR   SUPPLEMENTARY   READING 

King,  W.  I.     Elements  of  Statistical  Method,  Chapters  I— III. 
Rugg,  H.  O.     Statistical  Methods  Applied  to  Education,  Chapter  I. 
Thorndike,   E.   L.     Mental  and  Social  Measurements,   Introduction 
and  pages  36-41. 


CHAPTER   IV 

SCALES,  DISTRIBUTION  TABLES,  AND  SURFACES 
OF   FREQUENCY 

Thus  far  we  have  discussed  only  the  most  elementary 
statistical  matters.  But  we  have  seen  the  need  of  some 
technical  knowledge  of  statistics,  which  we  shall  now 
proceed  to  develop.  The  treatment  will  include  the 
meanings  of  the  various  statistical  terms,  methods  of 
calculating  them,  cautions  as  to  their  use,  and  devices  for 
showing  them  graphically. 

I.    SCALES 

Review  of  Scales.  The  first  essential  in  all  statistical 
work  is  to  determine  the  units  and  scales  to  be  used.  It  is 
impossible  to  collect  data  profitably  until  this  has  been 
done.  For  this  reason,  these  terms  were  discussed  at 
length  in  connection  with  the  collection  of  data.  If  the 
reader  is  not  familiar  with  the  treatment  given  there,1  he 
should  read  it  before  proceeding  with  thjs  section. 

For  our  purposes  here  it  is  necessary  to  understand  : 

1.  That  whenever  possible  all  the  measures  in  a  group  should  be 
expressed  in  terms  of  a  definite  common  measure  called  a  "unit." 

2.  That  all  measures  in  any  group,  when  arranged  in  order  of  size, 
make  a  scale  going  from  high  to  low. 

1  See  pp.  43-57 
100 


Scales  and  Distribution  Tables 


101 


M70 

160 
150 
140 
130 
IZO 
110 
100 


90 

70 

MS9  60 
50 

40 
30 


Q,47 


20 


University  High    169 


cr»»2«2£> 


Mishawaka  112 


Elgin  100  Maple    Lake,  Minn    100 


Granite    City  88 


East  Chicago    82 


..'South  Bend'62  Waukegan  63  - 
v  ||e  7^58  Rockford  59- 


Boone\ 


.     Brazil 56,Leavenworth  56. 
^^Nol^lie^i^Prgan  .ParK>3       G-ensbury 


54 


Norfolk, Neb.  42 

Bonner    Springs  38 
Junction  City,  Kan.   33 


Washington,  Mo.  41 

Russell,  Kan.  34 


Mt.  Carroll  30 


Fig.  6. 


Device  for  Representing  ;i  Discrete  Scale  (irapl 

tllCIliaticS    ill 


Thia^graph  shows  the  cost  per  1000  student    hours  in  i 
schools.         (From  J.  V.  Bobbin,  School  /I'tri'trc,  2'.i :  ."itllij 


ically. 
ci  rtain  hijxh 


102 


School  Statistics  and  Publicity 


3.  That  we  must  know  what  a  given  measure  on  a  scale  means, 
i.e.,  whether  6  extends  from  5.5  to  6.5  or  from  6.0  to  6.99  or  from  5.95 
to  6.05,  etc. 

4.  That  we  must  know  whether  the  scale  is  discrete  (all  measures 
separate  or  with  gaps  between)  or  continuous  (measures  running  into 
one  another  and  spread  out  all  over  the  scale). 

It  is  worth  noting  that  many  of  the  scales  the 
superintendent  uses,  especially  those  he  makes  up  for  him- 
self, are  discrete  in  appear- 
ance but  are  really  used  as 
though  they  were  continu- 
ous. That  is,  each  separate 
item  is  regarded  as  extend- 
ing half  the  distance  to 
the  nearest  items  above  and 
below. 

When  the  measures  are 
discrete  and  there  is  a  rela- 
tively small  number  of  cases, 
say  twenty  to  thirty,  they 
may  be  shown  in  a  Bobbitt 
table.1  In  this  kind  of  table 
the  name  of  each  measure 
is  written  in  the  left-hand 
column  and  the  size  of  the 
measure  in  the  right-hand 
column.  The  measures  begin 
with  the  highest  and  run 
to  the  lowest,  thus  giving 
an  idea  of  a  scale  like 
that  of  a  thermometer, 
which  goes  up  from  the 
bottom. 
18,  96 


REVERE- 2350 
QUINCY-  22.00 
CHELSEA- SI.  40 
ARUNGTON-20.80 
ICAMBRID0E- 20.40 
lMELROSE  -  20.  40 
EVERETT-  19.70 
MALDEN- 19.20 
WINTHROP- 19.20 
OMERVILLE-18.80 
ELMONT-  18.30 
WATERTOWN- 18.20 
MEDFORD-  18.00 
WINCHESTER-18.00 
NEWTON -17. 40 
DEDHAM-17.  40 
^BOSTON-  15.4-0. 
WALTHAM- 15.90 
BROOKLINE-  12.00 
MILTON- II.  50 


7.  —  Thermometer    Chart    for 
Presenting  a  Discrete  Scale. 

It  shows  the  total  tax  on  S1000,  for  the 
year  1912,  in  all  the  cities  and  towns  of 
the  metropolitan  district,  Boston,  Massa- 
chusetts.     Xote  omission  of  zero  line. 

(From  1012,  Newton,  Massachusetts, 
School  Report,  page  113.) 

1  See  pp. 


Scales  and  Distribution  Tables 


103 


Graphic  Presentation  of  Discrete  Scales.  A  discrete 
scale  is  easily  presented  graphically  by  various  methods. 
Four  of  these  methods  are  given  here,  three  of  them  using 
the  data  from  the  Bobbitt  table  on  page  17. 

1.  By  vertical  scale  on  left  with  names  of  items  to  the  right. 
Figure  6  shows  this. 


Cost  per  1000  5.  hours  $0 
Name  of  school. 
University  High  $169 

M'5hawaka,Ind.  M2 

Elgin,  III.  100 

Maple  Lake.Mmn  100 
Granite  City,  III. 

East  Chicago, Ind.  82 

Oe  Kolb,  III  74 

San  Antonio,  Tex  69 

Harvey,  III.  69 

Wa^egar.,111  63 

South  Bend.Ind.  62 

East  Aurora, III.  6 

Rockford,  III  59 

Boonev.lle.Mo  58 

Brazil,  Ind  56 

Leavenworth,  Kans.  5& 

Greensburqlnd  54 

Morgan  Pcrk.lll.  53 

Noblesville  Ind  52 

Norfolk  Neb-  42 

Washington,  Mo.  41 

Bonner  Spnnqs.KanS  38 

Russell,Kans.  34 

Junction  City. Kans.  33 

M+Carrolllll  30 


■so  ifec 


Fig.  8. 


Graphic  Representation  of  a  Bobbitt  Table,  Histogram  Form. 
The  data  are  from  Table  1,  page  18. 

2.  By  thermometer  device. 

Superintendent  Spaulding  in  his  1912  report"  for  the  schools  of 
Newton,  Massachusetts,  showed  a  similar  table  as  a  scale  on  a 
thermometer.  As  this  is  now  out  of  print,  the  device  is  here  re- 
produced. It  is  a  very  excellent  one,  save  that  the  zero  line  is  not 
properly  shown.     See  Figure  7. 


104 


School  Statistics  and  Publicity 


3.  By  bars  to  represent  magnitude. 

The  procedure  for  this  is :  First,  the  names  of  the  items  (schools 
in  this  instance)  are  placed  on  the  left,  the  highest  at  the  top.  In 
the  next  column  is  placed  the  corresponding  magnitude  (cost  per 
1000  student  hours  in  this  case).     At  the  top  and  running  out  to  the 


Cost  per  1000  S.  hrs    $0 


Name  of  school 

Umversiiy  Hi^ri 

$169 

MisriawaKa,  Ind- 

112 

Elq\r\,W. 

100 

Maple  Lake,  hinn. 

too 

Granite  City,  HI. 

88 

East  CrlicaOjO.lncl. 

82 

Oe  Kalb,  III. 

74 

San  Antonio, Tex. 

69 

Harvey,  HI. 

69 

WaukeojQnAW 

63 

Sooth  Bend,  Ind. 

bl 

South  Aurora,lU. 

61 

RocKford.lU. 

59 

Boorievvlle.Mo. 

58 

Braz.il,  Ind. 

56 

Leavenworth,  Kans 

56 

Greentouro],\noV 

54 

Morgan  ParKlW. 

53 

NobtesViUe,  Ind 

52 

Norfolk,  Neb. 

42 

Washington,  Mo. 

41 

Bonner  Spnnqb,  Kans.  3d 

RusseW.VSans. 

34 

Junction  City,  Kara. 

33 

MtCarroH.m. 

30 

Fro.  <).    -Graphic  Representation  of  a  Hoi  .hit  t  Table.  Smoothed  Curve  Form. 
The  data  arc  from  Table  1,  page .18.      This  is  simply  the  smoothed  form  of  Fig.  8. 


right,  the  scale  is  placed  (at  $10  intervals  here).  Each  distance  of 
five  small  squares  on  the  scale  represents  $10  and  one  small  square 
will  thus  mean  $2.  Therefore,  to  get  the  bar  for  the  first  school,  go 
out  on  the  scale  to  170,  drop  back  half  a  square  to  get  169,  and  then 
construct   the  bar  from  the  base  or  zero  line  to  this  point.     In  the 


Scales  and  Distribution  Tables  105 

same  way  the  bar  for  each  other  school  may  be  constructed.  The 
significance  of  the  bar  in  such  a  graph  lies  in  its  length  only,  not  in 
its  width.  Hence,  all  bars  in  the  same  diagram  must  be  uniform  in 
width.  The  bars  are  here  drawn  adjacent  to  each  other,  but  there 
can  be  spaces  between  if  desired.     See  Figure  8. 

4.  By  a  "  curve  "drawn  with  the  magnitudes  on  the  vertical 
axis  and  the  names  of  the  cases  running  in  order  from  high 
to  low  (or  vice  versa)  on  the  horizontal  axis. 

See  Figure  9. 

This  diagram  or  graph  has  been  constructed  in  exactly  the  same 
way  as  the  preceding  one,  except  that  dots  have  been  placed  at  the 
middle  of  the  spaces  where  the  ends  of  the  bars  came  in  the  preced- 
ing diagram,  and  then  joined  with  a  line.  The  dots  were  put  in 
faintly  and  have  been  covered  up  by  the  line.  Obviously,  then,  the 
"curve"  is  nothing  more  than  the  smoothing  down  of  the  corners 
of  the  bars. 

EXERCISES 

1.  The  salaries  of  school  superintendents  in  Missouri  cities  be- 
tween 2500  and  5000  population  in  1914-15,  for  all  cities  on  which 
data  could  be  obtained,  were  as  follows : 

Booneville,  $1,650  —  Butler,  $1,320  —  Cameron,  $1,400  —  Carter- 
ville,  $1,200  —  Caruthersville,  $1,200  —  Charleston,  $1,500  —  Clin- 
ton, $1,920  — De  Soto,  $1,400  —  Excelsior  Springs,  $1,500  Far- 
mington,  $1,400  —  Fayette,  $1,500  —  Festus,  $1,140  —  Frederick- 
town,  $1,450  —  Kennett,  $1,500  —  Kirkwood,  $2,400  -Liberty, 
$1,800  —  Louisiana,    $1,350  —  Macon,    $1,700  —  Marceline,    $1,100 

—  Marshall,  $2,100  —  Maryville,  $1,500  —  Monette,  $1,500     aUe" 
Hill,  $1,200  —  Richmond,  $1,500  —  Sikeston,  $1,500  —  Slater,  $1,200 

—  Warrensburg,  $1,600  —  Washington,  $1,170  — West  Plains,  $1,500. 
Make  up  a  Bobbitt  table  to  show  the  status  of  these  salaries,  and  graph 
this  table  in  as  many  ways  as  you  can. 

2.  Make  up  a  similar  table  on  superintendents'  salaries  in  a  group 
of  cities  in  some  other  state  that  may  be  legitimately  compared, 
choosing  cities  of  some  other  size  if  preferred. 

Statistics  on  population  may  be  gotten  from  census  reports  or  a 


106  School  Statistics  and  Publicity 

good  almanac  like  the  World  Almanac.  Figures  for  salaries  may  be 
gotten  from  directories  issued  by  book  companies,  from  reports  of 
the  state  superintendent  of  education,  or  from  "  A  Comparative  Study 
of  the  Salaries  of  Teachers  and  School  Officers"  (Bulletin  U.  S.  Bureau 
of  Education,  1915,  No.  31).  Volume  II  of  the  Annual  Report  of  the 
U.  S.  Commissioner  of  Education  will  furnish  figures  for  the  total  ex- 
pense of  the  superintendent's  office  but  sometimes  this  contains  other 
items  in  addition  to  his  salary.  In  the  smaller  cities  it  very  closely 
approximates  the  superintendent's  salary. 

II.    DISTRIBUTION   TABLES 

In  a  continuous  scale  or  distribution,  it  is  customary 
to  group  measures  of  magnitudes  that  are  almost  the 
same  in  value,  and  call  them  by  a  group  name.  Thus, 
in  the  Ayres  spelling  scale,  all  words  spelled  by  from  70 
to  76  per  cent  inclusive  of  children  in  a  given  grade 
are  lumped  into  one  group  and  called  73  in  difficulty  for 
that  grade. 

It  is  also  customary  in  such  a  distribution  to  make  up 
a  table  called  a  "  distribution  table  "  or  "  table  of  fre- 
quency." The  table  is  made  up  with  magnitudes  in 
order  of  size  in  the  left-hand  column  and  the  corresponding 
numbers  of  cases  or  frequencies  in  a  parallel  column  on 
the  right,  the  smaller  measures  preferably  being  at  the 
bottom.  The  grouping  is  for  the  purpose  of  condensation 
and  clearness,  but  the  cases  can  always  be  kept  individually 
^cessary.  Professor  Dearborn  in  a  study  of  grades 
has  kept  all  the  cases  of  each  magnitude^separate.  For 
example,  he  presents  the  distribution  of  marks  given  in 
English  to  69  eighth  grade  children  as  shown  in  Table 
13.1 

1  Dearborn,  W.  F. :  "  School  and  University  Grades,"  Bulletin, 
University  of  Wisconsin,  No.  368,  H.  S.  Series,  No.  9,  1910.  From 
Figure  9,  p.  25 


Scales  and  Distribution  Tables 


107 


Table  13.    Distribution  Table  of  Marks  of  Eighth  Grade 
Children  (from  Dearborn) 


Mark 

No.  Making 

Mark 

No.  Making 

100 

0 

80 

3 

99 

0 

79 

3 

98 

0 

78 

3 

97 

2 

77 

2 

96 

0 

76 

1 

95 

2 

75 

2 

94 

1 

74 

0 

93 

3 

73 

3 

92 

4 

72 

2 

91 

6 

71 

2 

90 

2 

70 

0 

89 

3 

69 

0 

88 

3 

68 

2 

87 

3 

67 

0 

86 

1 

66 

0 

85 

2 

65 

1 

84 

3 

64 

1 

83 

2 

63 

0 

82 

3 

62 

1 

81 

3 

61 

0 

60 

0 

The  magnitudes  in  this  table  run  from  60  to  97.  This 
table  is  exactly  like  a  Bobbitt  table,  except  that  in  this 
instance  there  are  several  cases  of  the  same  magnitude. 
But  even  here  these  cases  are  supposed  or  assumed  to  run 
from  the  lower  limit  of  the  magnitude  to  the  lower  limit 
of  the  next  magnitude.  That  is,  the  six  91's  are  not  all 
exactly  91  but  are  spread  from  barely'  91  up  to  not 
quite  92,  or  from  just  90.5  to  almost  91.5,  depending 
upon  which  system  of  definition  of  the  measure  91  is 
used.  ' 


108 


School  Statistics  and  Publicity 


But  we  know  that  such  grouping  as  Professor  Dearborn 
has  used  in  this  instance  is  rather  finer  than  we  are 
accustomed  to  use  in  examining  teachers'  marks. 

Let  us  now  make  coarser  groupings.  The  first 
possibility  that  occurs  to  the  reader  is  probably  that  of 
making  the  groupings  cover  supposedly  equal  parts  of 
the  scale,  say  five  units.  Thus,  we  may  make  the  groups 
cover  60-64,  65-69,  etc.,  setting  the  limits  very  definitely 
and  getting  these  groups : 


Marks 

No.  Making 

95-100 

4 

90-94 

16 

85-89 

12 

80-84 

14 

75-79 

11 

70-74 

7 

65-69 

3 

60-64 

2 

69 

Or  we  may  make  them  cover  60-69,  70-79,  etc.,  and 
have : 

Mark*  No.  Making 

90-100  20 

80-89  26 

70-79  18 

60-69  _5 

(J!) 

Or  we  may  group  as  many  school  system's  do,  and  have : 


Magnitude  ■ 

No.  Cases 

95  100 

4 

90-94 

16 

80  89 

26 

70-79 

18 

Below  70 

5 

69 

Scales  and  Distribution  Tables  109 

For  grouping,  this  from  Professor  Thorndike  should  be 
kept  in  mind : 

In  general,  in  mental  and  social  measurements,  in  the  calculation 
of  averages,  average  deviations,  and  mean  square  deviations,  when 
the  face  value  of  the  series  gives  a  grouping  of  40  to  60  steps,  it  is  al- 
lowable to  group  by  double  steps,  and  when  the  face  value  of  a  series 
gives  a  grouping  of  60  to  80  steps,  to  group  by  triple  steps.  But  it 
should  be  observed  that  coarse  grouping  saves  little  time  except  in 
the  calculation  of  the  average,  average  deviation,  and  mean  square 
deviation.  In  the  case  of  the  calculation  of  the  median,  25  percentile, 
75  percentile,  and  median  deviation,  it  is  the  author's  opinion  that 
the  gain  in  precision  from  the  finer  scale  is  greater  than  the  loss  in 
time,  if  one  economizes  time  in  recording  measures  in  the  finer  group- 
ing.1 

The  superintendent,  however,  may  not  understand  the 
finer  points  of  this  without  considerable  statistical  theory 
and  experience.  The  best  simple  rule  for  him  to  follow 
is  not  to  divide  into  small  groups  where  the  cases  seem  to 
bunch  more  closely  than  usual,  and  not  to  include  in  the 
same  group  cases  that  are  manifestly  far  apart.  Tn  the 
example  of  the  marks  given  above,  it  will  be  noted  that 
the  cases  bunch  together  very  closely  when  grouped  in 
the  third  form.  Nor  are  the  cases  to  be  found  in  one 
group  too  far  apart  from  one  another  to  be  in  that  group. 
For  instance,  81  and  89  are  to  be  found  in  the  same  group. 
But  experience  with  marks  shows  us  that  when  one  teacher 
marks  one  boy  81  and  another  teacher  marks  another 
boy  89  in  the  same  subject,  there  may  be  little  appreciable 
difference  in  the  achievements  of  the  boys. 

In  most  cases  common  sense  and  experience  must  be 
utilized  in  considering  grouping.  Salaries  of  superintend- 
ents may  be  safely  grouped   by  hundreds    (1200  1299, 

1300-1399,   etc.)   because  their  salary   increases  usually 
i 

1  Mental  and  Social  Measurements,  p.  50 


110  School  Statistics  and  Publicity 

come  by  hundreds.  But  the  salaries  of  grade  teachers 
are  more  profitably  grouped  by  fifties  or  twenty-fives, 
thus :  400-449,  450-499,  or  400-424,  425-449,  450-474, 
474-499,  etc.,  because  their  usual  salary  increases  are 
covered  by  the  smaller  steps.  A  grouping  for  training 
in  weeks,  of  teachers  that  covered  summer  school  work, 
would  be  1-6,  7-12,  13-18,  etc.,  or  1-3,  4-6,  7-9,  10-12 
and  not  1-4,  5-8,  9-12,  because  summer  schools  usually 
run  either  6,  9,  or  12  weeks. 

EXERCISES 

1.    The  salaries  of  superintendents  in  cities  of  2500-5000  for  1914- 
15,  for  all  obtainable,  were  in  the  states  given,  as  follows : 

Alabama.     $1,500  —  1,800  —  1,250  —  1,680  —  1,900  —  1,600  — 

1,800  —  1,500. 
Arkansas.     $2,000  —  1,620  —  1,000  —  1,500  —  1,600  —  1,600  — 

1,500  —  1,100  —  1,500  —  1,200  —  1,500  —  1,500  —  1,600  — 

1,350. 
Florida.     $1,500  —  1,200  —  1,500  —  1,650. 
Georgia.     $1,800  —  1,800  —  1,800  —  2,000  —  1,500  —  1,200  — 

1,500  — 1,500  — 1,200  — 1,600  —  1,800  —  2,000  —  2,000  —  1,650. 
Kentucky.     $1,000  —  1,500  —  1,800  —  1,350  —  1,800  —  1,600  - 

1,800  —  1,400  —  1,200  —  1,400  —  1,650  —  1,200  —  1,400  — 

1,500. 
Louisiana.     $1,500  —  1,800  —  1,500  —  1,800  —  1,500. 
Maryland.    $1,400  —  1,450. 

Mississippi.     $1,800  —  2,200  —  1,125  —  1,700  —  1,650  —  1,800. 
Missouri.     $1,650  —  1,320  —  1,400  —  1,200    ^1,200  —  1,500  — 

1,920  —  1,400  —  1,500  —  1,400  —  1,500  —  1,140  —  1,450  — 

1,500  —  2,400  —  1,800  —  1,350  —  1,700  —  1,100  —  2,100  — 

1,500  —  1,200  —  1,500  —  1,500  —  1,200  —  1,600  —  1,170  — 

1,500. 
North  Carolina.     $1,500  —  1,500  —  1,200  —  1,500  —  1,200. 
Oklahoma.     $1,500  —  1,500  —  1,400  —  1,800        1,800  —  1,500  — 

900  -  2,000    -  1,300        1,800    -  1,500  —  1,800  —  1,200  —  1.80C 

—  1,300  —  1,500  —  1,800  —  1,800. 


Scales  and  Distribution  Tables  111 

South  Carolina.  $1,200  —  1,500  —  1,215  —  1,500  —  1,350  — 
1,800  —  1,200  —  1,250  —  1,500  —  2,000. 

Tennessee.  $2,000  —  1,000  —  1,200  —  1,080  —  1,200  —  1,600  — 
1,600  —  1,500  —  1,000  —  1,800. 

Texas.  $1,960  —  2,000  —  1,500  —  1,800  —  2,100  —  2,200  — 
2,000  —  1,800  —  1,500  —  1,800  —  2,000  —  1,200  —  1,500  — 
1,560  —  1,800  —  1,500  —  1,800  —  1,300  —  1,800  —  2,300  — 
1,500  —  1,675  —  2,200  —  1,200  —  2,000  —  1,400  —  1,800  — 
1,500  —  1,500  —  1,500  —  1,800  —  1,800  —  1,800. 

Virginia.     $1,750  —  1,200  —  1,200. 

West  Virginia.     $1,500  —  1,400  —  1,350  —  1,500  —  1,500  —  1,800 

—  1,500  —  1,380  —  1,600  —  1,550  —  2080. 

Ignore  the  matter  of  sampling  and  arrange  these  salaries  in  a  distri- 
bution table,  being  careful  to  justify  the  step  chosen  in  your  grouping. 

2.  Make  a  similar  distribution  table  of  superintendents'  salaries 
in  cities  of  this  size  for  any  other  section  of  the  United  States,  getting 
your  data  from  the  sources  found  in  Exercise  2,  page  105,  and  telling 
just  why  you  use  each  step  in  the  process  of  making  the  table. 

3.  Make  a  similar  distribution  table  for  the  following  figures  on 
the  number  of  hours  required  by  individual  pupils  to  complete  one 
half-grade  in  grammar  :  : 

7  —  10  —  11  —  11  —  11  —  12  —  12  —  13  —  13  -^15  —  16  —  16 

—  16  —  17  —  18  —  18  —  19  —  19  —  20  —  20  —  21  —  21  —  22  — 
22  —  22  —  23  —  23  —  25  —  27  —  29  —  33  —  33  —  33  —  34  —  34 

—  36  —  37  —  38  —  39  —  40  —  43  —  44  —  44  —  48  —  49  —  49. 

III.    SURFACE    OF   FREQUENCY 

Graphing  Distribution  Tables.  For  the  presentation  of 
grouped  distributions  by  graphs,  three  simple  devices 
are  available,  the  histogram  or  rectangular  graph,  the 
smoothed  graph,  and  the  check  form  of  the  histogram. 

The  procedure  for  this  histogram  is  as  follows : 

1.  Lay  off  on  cross-section  paper  a  horizontal- scale,  on  which  the 
magnitude  scale  runs  by  groups  from  the  lowest  magnitude  at  the  left 
to  the  highest  magnitude  at  the  right. 

1  Fr6m  Monograph  C,  Individual  Instruction,  San  Francisco  State 
Normal  School,  p.  28 


112 


School  Statistics  and  Publicity 


2.    From  the  same  zero  point  erect  a  perpendicular  scale  which 
is  to  represent  the  number  of  cases. 


' 

25 

20 

) 

> 

15 

- 

r        I 

1 

10 

5  ; 

i 
i 
1 

1 

- — 

- 

r 

i 
1 

I 

l 

i 

No       below 
Cases         70 


10  1 079       80  to  89      90  to  84       95  to  WO 


Ik..    10.  —  Histogram  Showing  Data  of  Table  l'A,  but   Using  Grouping  as 
Given  on  Page  108. 


•'5.  Then  find  on  the  horizontal  scale  the  point  marking  the  mag- 
nitude of  any  given  case,  and  count  up  to  find  the  proper  point  to 
denote  the  number  of  cases  in  that  group. 


Scales  and  Distribution  Tables 


113 


Do  the  same  for  each  group.  In  so  doing,  one  will  get  a  number  of 
points  at  different  heights  strung  out  above  the  horizontal  scale. 

4.  Then  proceed  to  draw  a  line  through  these  points  coming  down 
to  the  base  line  on  the  right,  and  either  coming  down  to  the  base  line 


25 


20 


(5 


10 




\ 

— 

— 

■ 

\ 

\ 

* 

No.      Bolov 
Cases         70 
Fig.   11. 


70  to  79      ©0  to  89     90  to  94       95  folOO 
Smootlicd  Form  of  Graph  Shown  in  Figure  10. 


on  the  left  or  going  to  the  vortical  scale  on  thai  side.     There  will  thus 
be  incl6sed  an  area  which  is  called  the  "surface  of  frequency. 

This  surface  may  be  made  by  making  each  point  located  after  the 


114  School  Statistics  and  Publicity 

manner  described  above,  the  upper  left-hand  corner  of  a  rectangle 
which  is  as  wide  as  the  length  of  the  space  occupied  by  that  group  on 
the  horizontal  scale.  Thus  the  plotting  of  the  Dearborn  data  as  a 
histogram  is  shown  in  Figure  10. 

It  is  not  customary  to  draw  those  parts  of  each  rectangle 
shared  in  common  with  other  rectangles.  Common  por- 
tions in  the  diagram  are  shown  by  dotted  lines. 

In  the  smoothed  graph  (Figure  11),  the  points  located 
to  determine  the  rectangles  may  represent  the  middles 
of  the  tops  of  such  rectangles  instead  of  the  upper  left- 
hand  corners.  Then  these  points  maybe  joined  by  straight 
lines,  giving  a  surface  with  apexes.  This  is  somewhat 
"  smoothed,"  it  will  be  noticed. 

The  check  form  of  the  histogram  simply  uses  dots  on 
cross-section  paper,  one  for  each  item,  thus  keeping  the 
columns  the  same  width.  There  is  no  line  drawn  above, 
the  columns  showing  roughly  the  shape  of  the  surface. 
It  is  a  very  valuable  form  for  tabulating  data  and  at 
the  same  time  showing  the  shape  of  the  surface  of  fre- 
quency. That  is,  it  may  be  made  up  before  the  dis- 
tribution table.  Thus,  if  cross-section  paper  had  been 
used  for  the  Dearborn  data  at  the  outset,  a  surface  of 
frequency  like  that  in  Figure  12  could  have  been  obtained, 
and  from  this  surface,  the  distribution  table  given  on  page 
107  could  have  been  easily  made  up. 

In  graphing  a  distribution  table,  the  seaje  in  which  the 
items  come  regularly  is  always  put  on  the  base  line. 
The  scale  in  which  the  items  come  more  or  less  irregularly 
is  put  on  the  vertical  line.  In  other  words,  this  means 
that  magnitude  is  measured  on  the  horizontal  line  of  the 
graph  and  the  number  of  cases  is  shown  on  the  vertical 
line.  There  is  no  bullet-proof  reason  for  this  ;  it  is  simply 
the  conventional  way  of  doing  the  thing,  just  as  the  order 


Scales  and  Distribution  Tables 


115 


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1 16  School  Statistics  and  Publicity 

from  left  to  right  across  the  page  and  from  top  to  bottom 
in  reading  and  writing  is  the  proper  and  customary  one 
for  European  people  and  their  descendants.  One  should 
draw  pictures  or  graphs  of  data  to  be  read  with  as  much 
care  as  he  would  exercise  in  preparing  the  manuscript  of 
an  article  to  be  printed.1 

Characteristics  of  Surface  of  Frequency.  The  dis- 
tribution table  is  a  great  economy  over  a  miscellaneous 
mass  of  unassorted  data.  But  it  is  too  cumbersome  to 
be  kept  in  mind  in  all  its  details.  We  need  to  apply  here 
our  idea  of  the  fundamental  image  or  bird's-eye  view  of 
the  whole.  This  can  best  be  done  through  the  use  of 
two  special  qualifications,  characteristics,  or  earmarks  of 
the  distribution.  The  first  of  these  qualities  is  that 
measure  which  indicates  the  typical,  average,  or  central 
size  of  the  group.  The  second  is  that  number  which 
indicates  how  far  the. other  members  of  the  group  on  the 
average  vary,  spread,  or  deviate  from  the  first-named 
quality.  These  two  characteristics  of  the  distribution 
table  serve  to  make  it  full  of  meaning  to  any  person  who 
understands  statistics ;  and  with  a  little  care  the  lay 
reader  may  readily  acquire,  the  ideas  back  of  these 
devices. 

Both  of  these  characteristics  may  be  expressed  as  magni- 
tudes or  as  so  many  multiples  of  whatever  unit  may  be 
used  in  the  distribution  table.  They  may  also  be  shown 
graphically  on  the  surface  of  frequency.  The  particular 
kind  of  central  tendency  to  be  used  and  the  measure  from 
it  depend  altogether  upon  the  shape  of  this  surface  of 
frequency.  The  question  at  once  comes  up,  then : 
How  many  variations  in  the  surface  of  frequency  are  of 

1  Paraphrased  from  W.  C.  Brinton :  Graphic  Methods  of  Present- 
ing Facts 


Scales  and  Distribution  Tables  117 

significance  to  the  superintendent,  and  how  may  he  know 
them? 

Normal  Surface  of  Frequency.  If  the  cases  in  any 
distribution  are  taken  by  chance  or  by  a  combination  of 
causes  that  amount  to  chance,  the  shape  of  the  surface 
of  frequency  invariably  becomes  bell-shaped  with  symmet- 

Percerrr 
maKtrig 
given  score 

20 


10    - 


Lowest  Highest 

Score  score 

Fig.  13.  —  Example  of  Normal  Surface  of  Frequency.  This  graph  shows 
the  per  cent  of  pupils  attaining  given  scores  in  Stone  Reasoning  Tests,  all 
pupils  being  tested.      (Adapted  from  Butte  Survey,  page  95.) 

rical  sides.  Figures  13,  14,  and  15  are  good  examples 
because  all  the  children  were  taken,  and  the  variations 
then  arise  only  from  chance. 

Notice  that  in  one  of  these  graphs  or  diagrams  the  cases 
composing  it  are  bunched  much  more  closely  around  the 
highest  point  or  apex  of  the  distribution  than  in  the 
second  diagram.  But  it  might  sometimes  happen  that 
two  different  sets  of  children,  when  tested  on  the  same 
thing,  might  give  distribution  tables  which,  when  graphed, 
would  make  graphs  or  surfaces  of  frequency  varying  as 
widely  as  these  two.  For  example  two  sixth  grades 
might  make  the  same  average  or  central  tendency  on  the 
Courtis  Tests.     But  in  one  case  the  achievements  of  all 


118 


School  Statistics  and  Publicity 


the  children  might  be  close  to  the  average,  while  in  the 
second  case  some  of  the  children  might  make  very  high 
records  and  others  very  low.  This  bell-shaped  surface  is 
called  the  "  normal  "  or  "  probability  "  surface  because 
it  is  the  one  found  in  natural  and  mental  phenomena  of 
all  kinds,  when  the  distributions  are  made  up  from  un- 


5043 


4462 


3536 


2878 


1373 


462 


Z     59 


3536 


1884 


1225 


527 


399 


0 

10 

20 

30 

40 

50 

60 

70 

80 

90 

100 

110 

120 

to 

to 

to 

to 

to 

to 

to 

to 

to 

to 

to 

to 

to 

9 

19 

29 

39 

49 

59 

69 

79 

69 

99 

109 

119 

129 

Fi<;.  14.  -Example  of  Normal  Surface  of  Frequency.  This  figure  shows 
the  number  of  pupils  writing  at  each  speed  from  0  to  9  letters  per  minute 
to  120  to  129  letters  per  minute.  Data  for  25,387  pupils  in  four  upper 
grades  of  Cleveland.  (From  Measuring  the  Work  of  Public  Schools,  Cleve- 
land Survey,  page  ^37,  by  permission.) 


selected  cases  or  from  those  picked  at  random.  In  other 
words,  it  is  the  normal  or  probable  thing  to  expect  from 
such  data. 

Skew  Surface  of  Frequency.  But  there  are  some 
distributions  in  which  the  cases  bunch  much  more  on  one 
side  of  the  apex  than  on  the  other,  for  the  reason  that  a 
certain  cause  or  combination  of  causes  operates  on  some 
of  the  cases  in  the  distribution  but  not  on  all.     The 


Scales  and  Distribution  Tables  119 

distribution  or  surface  is  said  to  be  "  skewed  "  toward 
the  thin,  drawn-out  side  and  is  called  a  "  skew  "  dis- 
tribution or  "  skew  "  surface.1  A  good  example  of  such 
a  distribution  is  one  made  up  of  the  number  of  children 
of  different  ages  in  school.  As  the  children  grow  older, 
some  die,  and  consequently  the  group  of  a  given  age  is 


5000- - 


Fig.   15.  —  Smoothed  Form  of  Figure  14. 

slightly  smaller  in  size  than  any  group  of  lower  age.  But 
at  the  age  of  fourteen  the  compulsory  education  laws 
usually  cease  to  operate,  and  many  children  immediately 
drop  out  of  school.  Others  want  to  go  to  work.  Such 
forces  tend  to  decrease  the  group  rapidly.  The  follow- 
ing table  of  frequency  for  1908  for  Nashville,  Tennessee, 
as  reported  by  Strayer  in  his  "  Age  and  Grade  Census  of 

1  From  the  verb  "skew,"  meaning  to  put  askew  or  twist  to  one  side. 
While  there  is  perfect  agreement  among  educational  writers  as  to  what 
constitutes  a  skew  distribution,  there  is  no  agreement  as  to  which  is 
the  skew  end.  By  this  term  some  writers  mean  the  blunt  end  and 
some  the  sharp  end.  The  author  uses  it  to  describe  the  sharp  end, 
following  Professor  Thorndike. 


120  School  Statistics  and  Publicity 

Schools  and  Colleges,"  1  well  illustrates  the  point  under 
discussion : 

Table  14.     Distribution  by  Ages  of  Pupils  in  the  Nashville 
Schools,  1908 


Age 

Number  in  group 

72 

1493 

8 

1733 

9 

1584 

10 

1712 

11 

1595 

12 

1626 

13 

1490 

14 

1198 

15 

885 

16 

528 

17  and  over 

390 

Notice  that  there  is  practically  no  increase  or  decrease 
of  importance  in  these  groups  until  the  age  of  fourteen 
is  reached.  From  this  point  on,  the  decrease  is  very 
rapid.  If  the  ages  beyond  seventeen  had  been  given  sep- 
arately, the  extension  would  narrow  down  to  a  very 
slender  one.  Figure  16  shows  the  facts  in  the  histogram 
or  unsmoothed  form. 

Smoothed  out,  as  in  Figure  17,  the  g«aph  shows  the 
"  skew  "  even  better. 

A  good  example  of  a  skew  surface  is  shown  in  the  results 
of  the  spelling  test  in  Figure  18. 

1  Bulletin  U.  S.  Bureau  of  Education,  1911,  No.  5,  p.  34 

2  Children  are  not  permitted  to  enter  the  Nashville  schools  until 
they  are  seven  years  of  age. 


Scales  and  Distribution  Tables 


121 


Mo.  of 

pupil5 

1600 

1600 

1400 
(200 
1000 
800 
600 

400 

200 

Ages      6       7       8       9      10      II       12      13      W      15      16      17  and  over 

Fig.    16.  —  Skewed    Histogram    Representing    Distribution    of    Pupils    by 
Ages  in  Nashville  Public  Schools,  1908.      (From  data  in  Table  11.) 

Mo.  of 
pupils 
1800 

1600   - 


1400 
1200 

1000 
800 
600 
400 
200 


Ages     6 
Fn 


/ 

■^ 

/ 

— 

/  1 

1      "T 

|\ 
!\ 

. ( — 

\ 

\  i-    - 

_J 

\ 

— 

1 

I        9       10       II       '2       13       14       15       16      17  and  over 
Smoothed  Form  of  Craph  (liven  in  Figure  10. 


122  School  Statistics  and  Publicity 

#  Of  children  M9i 

40  H 


30- 


20- 


10- 


ENTIRE  CITY 
3988  children 


10    20  30  40  50  60  70  80  90  100 

Fig.  IS.  —  Skewed  Histogram  Showing  the  Percentage  of  Children 
Attaining  Each  of  the  Possible  Scores  on  the  Spelling  Test  in  Salt  Lake  City 
as  a  Whole.      (Adapted  from  Salt  Lake  City  Survey,  page  135.) 

EXERCISES 

1.  Draw  a  surface  of  frequency  for  each  of  the  distribution  tables 
used  or  gotten  up  by  you  in  the  previous  exercises. 

2.  Draw  a  surface  of  frequency  for  each  of  th<J»<distributions  given 
in  the  table  on  page  123. 

REFERENCES  FOR  SUPPLEMENTARY  READING 

King,  W.  I.     Elements  of  Statistical  Method,  Chapters  V,  XL 
Rugg,  H.   O.     Statistical  Methods  Applied  to  Education,     Chapters 

IV,  VII,  VIII. 
Thorndike,  E.  L.     Mental  and  Social  Measurements,  Chapter  II  and 

pages  28-36. 


Scales  and  Distribution  Tables 


123 


Frequency  of  the  Different  Percentages  of  Boys  and 

Girls  Retarded  Two  Years  in  Certain  Cities  of  25,000 

Population  and  Over,  1908  1 


Per  cent  of  total 
no.  of  boys 

No.  cities 

Per  cent  of  total 
no.  of  girls 

No.  cities 

2 

1 

2 

3 

3 

5 

3 

6 

4 

3 

4 

9 

5 

7 

5 

9 

6 

7 

6 

9 

7 

4 

7 

12 

8 

9 

8 

18 

9 

16 

9 

15 

10 

12     64 

10 

13 

11 

12 

11 

11 

12 

19     56 

12 

7 

13 

11 

13 

3 

14 

9 

14 

2 

15 

2 

15 

5 

16 

2 

16 

3 

17 

1 

17 

3 

18 

7 

18 

2 

19 

1 

19 

2 

20 

2 

21 

1 

21 

1 

22 

1 

1  From  Strayer,  G.  D.  :    "Age  and  Grade  Census  of  Schools  and 
Colleges,"  Bulletin  U.  S.  Bureau  of  Education,  1911,  No.  5,  pp.  86-87 


CHAPTER  V 

MEASURES    OF   TYPE 

There  are  three  measures  of  the  "  type  "  or  central 
tendency  of  importance  in  school  work  —  the  mode,  the 
median,  and  the  average  —  which  will  now  be  discussed 
in  order. 

I.   THE   MODE 

Definition.  The  mode  is  that  number  which  represents 
the  size  of  the  most  numerous  item  or  items  in  a  group. 
That  is,  it  is  the  vogue  or  fashion  in  the  cases,  because 
there  are  more  of  this  size  than  of  any  other.  The  mode 
is  precisely  what  the  ordinary  man  usually  has  in  mind 
when  he  speaks  of  the  "  average."  He  is  referring  to  that 
measure  which  includes  the  greatest  number  of  cases.  If 
he  says  that  teachers  instruct  forty  children  on  the 
average,  he  means  that  more  teachers  instruct  just  about 
forty  children  than  teach  thirty,  fifty,  or  any  number 
far  removed  from  forty. 

Graphic  Representation.  Graphically,  the  mode  is 
the  magnitude  represented  by  the  point  on  the  scale 
above  which  the  surface  of  frequency  is  highest.  It  may 
be  marked  by  a  perpendicular  erected  from  this  point 
to  the  apex  of  the  surface.  But  note  that  the  mode  is  a 
measure,  not  a  number  of  cases. 

Calculation.  All  that  is  necessary  here  is  to  pick  out 
the  group  containing  the  largest  number  of  cases  and 

124 


Measures  of  Type 


125 


see  what  its  magnitude  is.  If  several  adjoining  groups 
have  about  the  same  number  of  cases  in  them,  they 
should  be  run  together  into  larger  groups.  Usually  this 
procedure  will  give  a  more  pronounced  mode,  which  is 
its  purpose.  In  case  two  widely  separated  groups  are 
larger  than  the  others,  the  distribution  is  said  to  be  "  bi- 


Number  receiving 
125 


Salary 


oo  o  ooooo  Oo  o  o  °  o  o 
loO  incooOJLOco  <5o  m  ir>  <->>  0  oj 
<o   r-    r^t^oooooo<o<s>cicr>cncr>oo 


Fig.    19.  —  Example  of  Multi-modal  Surface  of  Frequency. 
This  shows  the  distribution  of  salaries  paid  elementary  school  teachers,  Salt    Lake 
City,  1914-15.      Note  that  the  printing  on  this  graph  cannot  be  read  easily  from  one 
position.      (Adapted  from  Halt  Lake  Vila  Survey,  page  51.) 

modal,"  that  is,  it  has  two  modes,  and  probably  is 
composed  of  two  rather  different  classes  which  have  been 
lumped  together  but  which  possibly  should  not  be  so 
considered.  A  frequency  table  showing  the  number  of 
teachers  getting  different  salaries  will  often  show  more 
than  one  mode.     For  example,  the  surface  of  frequency 


126  School  Statistics  and  Publicity 

for  the  salaries  of  elementary  teachers  in  the  Salt  Lake 
City  Survey  has  three  well-defined  modes  at  approximately 
$650,  $850,  and  $1020.  This  means  that,  as  regards 
salaries,  there  are  really  three  distinct  classes  of  teachers 
within  the  whole  group.  The  graph  is  here  shown  in 
Figure  19. 

Advantages  of  the  Mode  for  School  Statistics.  1.  The 
mode  is  useful  where  it  is  desirable  to  eliminate  extreme 
variations. 

For  example,  the  amount  of  work  a  given  group  of  children  can  do 
in  a  school  year  is  determined  by  the  modal  attendance  of  the  group, 
not  by  that  of  the  few  who  are  absent  almost  continuously,  or  that 
of  the  small  number  who  never  miss  a  day. 

2.  In  finding  the  mode,  it  is  unnecessary  to  know 
anything  about  extreme  cases  except  that  they  are  few 
in  number. 

In  comparing  his  school  with  other  schools,  the  superintendent 
need  not  worry  about  the  one  or  two  schools  that  are  higher  than  any 
of  the  others,  if  his  own  school  falls  in  the  mode  or  close  to  it.  The 
extreme  cases  may  not  be  measured  accurately,  and  they  may  or  may 
not  really  come  legitimately  into  the  distribution.  But  whether 
they  do  or  not,  they  cannot  affect  the  mode. 

3.  It  is  very  easy  to  determine  with  considerable 
accuracy  from  well-selected  data. 

4.  It  is  the  best  measure  of  type  to  the  ordinary  mind. 

As  before  indicated,  this  is  what  the  ordinary  man  often  means  by 
"average." 

5.  It  is  unambiguous. 

No  one  ever  thinks  from  it  that  all  the  measures  in  the  group  are 
practically  on  it. 

6.  The  mode  is  often  the  most  typical  measure  of  a 
skew  distribution. 


Measures  of  Type  127 

Probably  the  most  significant  thing  about  a  frequency  table  of 
teachers'  salaries  is  that  largest  group  which  get  the  same  salary  or 
a  salary  within  certain  limits.  The  extreme  salaries,  the  median 
salary,  or  the  average  salary  might  be  of  no  especial  significance.  But 
the  modal  salary  would  point  out  the  significant  group  at  once.  The 
three  modal  salaries  for  the  Salt  Lake  City  elementary  teachers,  as 
shown  in  the  graph  on  page  125,  indicate  the  significant  salaries  at  a 
glance. 

Disadvantages  of  the  Mode  for  School  Statistics.  1.  In 
many  groups,  no  single,  well-defined  type  actually  exists. 

There  is  no  such  thing  as  a  modal  age  for  children  or  a  modal  num- 
ber of  children  in  a  grade  in  school.  All  the  age  groups  up  to  14  are 
about  the  same  in  size,  and  all  the  grades  up  to  about  the  sixth  or 
seventh  keep  about  the  same  size.  Of  course  children  drop  out  but 
usually  enough  are  held  over  to  make  the  grades  approximately  the 
same  size. 

When  all  cases  are  kept  separate  as  in  Bobbitt  tables,  there  is,  of 
course,  no  mode  unless  several  cases  happen  to  be  of  exactly  the  same 
size  or  are  considered  to  be  of  the  same  size. 

2.  The  mode  is  of  no  value  if  weight  is  to  be  given  to 
extreme  cases. 

It  would  take  no  special  account  of  the  high  per  capita  cost  of  a 
city  at  the  upper  end  of  a  per  capita  group,  so  far  as  the  size  of  the 
item  was  concerned,  although  such  city  might  admittedly  have  the 
best  schools  in  the  group.  Similarly  it  would  take  no  special  note  of 
the  lowest  city  in  the  group  although  it  might  admittedly  have  the 
worst  schools  in  the  group. 

3.  The  mode  cannot  be  determined  by  any  simple 
arithmetical  process  and  is  sometimes  difficult  to  get  by 
any  method. 

4.  The  product  of  the  mode  by  the  number  of  items 
does  not  give  the  correct  total. 

For  example,  take  Table  15. 


128  School  Statistics  and  Publicity 

Table  15.    Distribution  Table  Showing  Penmanship 
Records  of  Second  Grade  at  Butte  1 

Score  Number  pupils  making 

4  .  5 

5  22 

6  21 

7  29 

8  28 

9  42 

10  7 

11  29 

12  5 

13  7 
16                      _1 

196 

The  mode  in  this  example  is  9.  196  X  9  =  1764.  The  sum  of 
the  products  of  the  cases  in  each  group  by  its  score,  however,  is  only 
1617.  Such  a  total  sometimes  proves  very  useful  for  checking  other 
steps. 

5.  The  mode  may  be  determined  by  a  very  few  items 
in  case  none  of  the  groups  contains  more  than  a  few 
items. 

This,  of  course,  may  be  offset  by  wider  grouping. 

EXERCISES 

1.  What  is  the  mode  in  each  of  the  distribution  tables  used  in  pre- 
vious exercises? 

2.  Draw  the  line  to  represent  the  position  of  tl^mode  on  each  of 
the  surfaces  of  frequency  used  in  previous  exercises. 

II.    THE   MEDIAN 

Definition.     The  median  is  the  magnitude  represented 
by  the  mid-point  on  a  scale  or  distribution.     Obviously, 
half  the  cases  fall  below  this  mid-point  and  half  above  it. 
1  Butte  Survey,  p.  80 


Measures  of  Type  129 

Note  that  the  median  is  a  magnitude  or  size  of  a  case, 
not  the  number  of  the  case. 

Calculation.  Various  devices  and  formulas  have  been 
given  for  calculating  the  median,  according  as  there  is  an 
odd  number  of  cases,  an  even  number  of  cases,  a  gap 
between  two  groups  that  are  equal  in  size,  etc.  But  the 
simplest  and  surest  plan  by  far  is  to  regard  the  distribution 
as  a  scale  and  always  to  find  the  magnitude  of  the  mid-point 
on  it. 

If  there  is  an  odd  number  of  cases,  the  median  is,  of 
course,  where  the  mid-point  of  the  middle  case  lies.  If 
the  median  falls  in  a  gap,  the  mid-point  in  the  gap  must 
be  taken.  If  the  median  falls  in  a  group  distributed 
over  part  of  the  scale,  one  must  run  up  the  part  covered 
by  this  group  until  he  finds  the  point  that  will  exactly 
place  half  the  cases  in  the  whole  distribution  below  it 
and  half  above  it,  splitting  a  case  into  halves  if  it  is 
necessary. 

The  essential  thing  is  to  find  the  mid-point.  The  mag- 
nitude denoted  by  this  mid-point  will  be  the  exact  median. 
The  magnitude  corresponding  to  the  group  containing  the 
mid-point  will  be  the  approximate  median.  This  mid- 
point method  of  calculation  will  now  be  illustrated  with 
various  examples,  starting  with  an  even  number  of  cases. 

If  the  distribution  is  a  discrete  one  and  contains  an 
even  number  of  cases,  the  median  falls  between  the  two 
middle  cases.  The  place  for  it  to  fall  is  found  by  dividing 
the  number  of  cases  by  2,  which  gives  the  number  of 
cases  to  have  on  one  side.  Then  count  in  from  one  end 
till  this  number  of  cases  has  been  checked  off. 

Fc)r  instance,  suppose  we  make  up  a  Bobbin  table  with  an  even 
number  of  cases  by  taking  only  the  first  twelve  cases  from  the  table 
on  page  18,  as  in  Table  16. 


130 


School  Statistics  and  Publicity 


Table  16.    Bobbitt  Table  Showing  Cost  of  Instruction  per 
1000  Student  Hours  (Mathematics) 


Name  of  school 


University  High 
Mishawaka,  Ind.     . 

Elgin,  111 

Maple  Lake,  Minn. 
Granite  City,  111.  . 
East  Chicago,  Ind. 
De  Kalb,  111.  7~. 
San  Antonio,  Tex.  . 
Harvey,  111.  .  .  . 
Waukegan,  111.  .  . 
South  Bend,  Ind.  , 
East  Aurora,  111. 


Cost  per  1000 
student  hours 


$169 
112 
100 
100 

88 
82 


74 
69 
69 
63 
62 
61 


The  mid-point  of  these  twelve  cases  will  obviously  have  to  throw  six 
cases  on  each  side  of  it,  that  is,  it  must  come  between  cases  six  and 
seven.  For  those  who  desire  a  formula,  this  will  be  found  by  dividing 
the  number  of  cases  by  2.  The  mid-point,  then,  is  at  the  magnitude 
halfway  between  $82  and  $74  or  at  $78.  ($82  -  $74  =  $8.  \  of  $8 
=  $4.  $74  +  $4  =  $78.)  The  same  result,  of  course,  could  be 
obtained  by  merely  taking  the  average  of  the  two  middle  cases. 
($74  +  $82  =  $156.     $156  -^  2  =  $78.) 

If  the  two  cases  between  which  the  median  in  a  Bobbitt  table  falls 
are  the  same  size,  as  in  the  Bobbitt  table  of  size  of  classes  on  page  96, 
the  median,  of  course,  is  represented  by  the  size  of  e^£her  case,  —  19  in 
this  instance. 

For  an  example  of  a  continuous  series  and  even  number 
of  cases,  take  the  achievements  of  the  eighth  grade  in 
composition  at  Butte.1 

These  may  be  adapted  for  our  purposes  as  follows,  paying  attention 
for  the  present  to  the  two  left-hand  columns  only: 
1  Bvftc  Survey,  p.  74 


Measures  of  Type  131 

Rated  at  Number  of  papers 

7  2 

6  6         (8)    Adding  down 

5  22         (30) 

4  43         (73) 

3  39 

2  32         (42) 

1  9         (10)  Adding  up 

0  1 

2)154 
77 

There  are  154  cases  in  all.  The  median,  then,  must  be  at  the  point 
which  will  throw  77  cases  on  one  side  and  77  cases  on  the  other.  This 
point  will  obviously  be  at  the  end  of  the  77th  case  or  the  beginning  of 
the  78th  case,  as  one  prefers  to  call  it.  Say  that  this  point  will  be 
located  at  the  end  of  the  77th  case.  Counting  up  we  find  42  cases  in 
steps  0,  1,  and  2.  If  the  39  cases  in  step  3  are  added,  we  find  that  we 
have  more  than  the  required  77  cases.  Subtracting  42  from  77  we 
find  that  we  must  have  35  cases  more.  That  is,  we  must  add  35  cases 
from  group  3  to  the  other  42  so  as  to  reach  the  end  of  the  77th  case. 
This  means  that  the  median  is  located  §§  of  the  distance  up  the  scale 
represented  by  the  step  3.     (§§  =  .90) 

But  does  step  3  extend  from  3  to  4,  or  from  2.5  to  3.5?  If  it  ex- 
tends from  3  to  4,  the  median  is  obviously  3.90.  (3  +  .90  =  3.90) 
In  the  Butte  scoring,  however,  the  latter  method  was  used,  and  the 
actual  values  on  the  Hillegas  scale  are  as  follows : 

0  is  0  3  is  3.69 

1  is  1.83  4  is  4.74 

2  is  2.60  5  is  5.85,  etc. 

With  this  in  mind,  we  must  use  the  true  measures  on  the  Hillegas 
scale.  Step  3  would  extend  from  the  halfway  point  between  2.60 
and  3.69  (or  3.145)  and  the  mid-point  between  3.69  and  4.74  (or 
4.215).  The  distance  between  3.145  and  4.215  is  1.07.  \\  of  1.07 
is  .96.  Adding  this  .96  to  3.145  we  get  the  median  or  mid-point,  4.105. 
The  median  may  also  be  figured  from  the  other  end.  The  pro- 
cedure is  the  same,  the  figures  this  time  being  as  follows,  ('minting 
down  we  find  that  from  quality  4  up  we  have  73  cases.  We  need  to 
take ^4  cases  from  the  upper  part  of  the  group  rated  3.  That  is.  we 
must  go  down  s%  of  the  part  covered  by  that  group.      (&        .10)    If 


132  School  Statistics  and  Publicity 

the  step  meant  3  to  4,  the  median  would  then  be  4  —  .10  or  3.9.  But 
as  we  saw  before,  this  group  covers  1.07  and  extends  up  to  4.215. 
s%  of  1.07  is  .11.     Then  4.215  -  .11  gives  4.105  for  the  median. 

If  the  distribution  is  a  discrete  one  and  contains  an  odd 
number  of  cases,  the  magnitude  of  the  middle  case  is  the 
median.  This  is  easily  found  by  counting  in,  usually 
adding  1  to  the  number  of  cases  and  dividing  by  2  to 
get  the  number  of  the  middle  case. 

Thus,  in  the  discrete  series  represented  by  the  first  two  columns 
of  the  table  on  page  51,  the  median  real  wealth  behind  each  $1  for 
schools  is  $234,  because  it  is  the  eighteenth  case  on  the  scale.  There 
are  17  cases  below  and  17  cases  above.  Note  that  18,  the  number 
of  case  wanted,  is  found  by  adding  1  to  the  total  number  of  cases  (35) 
and  dividing  by  2.  (1  +  35  h-  2  =  18.)  In  the  Bobbitt  table  on  page 
18,  the  median  is  represented  by  the  thirteenth  case.  (13  =  1+25  -i-2.) 
Note  the  horizontal  lines  inclosing  the  median. 

If,  however,  the  series  is  a  continuous  one  and  has  an 
odd  number  of  cases,  the  median  is  manifestly  located  at 
the  mid-point  of  the  middle  case.  In  the  discrete  series, 
we  took  the  whole  middle  case  for  the  median.  In  the 
continuous  series,  the  middle  case  is  itself  supposed  to  be 
spread  out  along  the  scale,  and  consequently  we  have  to 
find  its  mid-point. 

For  example,  let  us  take  the  median  for  the  fifth  grade  composi- 
tion scores  at  Butte.1     These  were  as  follows : 


Rated  at 

Number  mak 

ing 

5 
4 

1 

18 

(19)  Adding  down 

3 

49 

(68) 

2 

86 

1 

46 

(47)  Adding  up 

0 

1 

2)201 

100.5 
1  Butte  Survey,  page  74 


Measures  of  Type  133 

This  time  we  have  an  odd  number  of  cases.  The  median  will  fall 
on  that  point  where  100}  cases  come  on  either  side,  that  is,  in  the 
middle  of  the  101st  case.  In  steps  0  and  1,  we  have  47  cases.  We 
need  53  V  cases  out  of  the  86  in  step  2  to  find  our  halfway  place.  Chang- 
ing step  2  to  the  Hillegas  value  as  before,  we  find  that  it  extends  from 
the  mid-point  between  1.83  and  2.60  to  the  mid-point  between  2.60 

and  3.69,  that  is,  from  2.215  to  3.145,  the  distance  being  .93.     58'5 

86 
of  .93  is  .58.     2.215  plus  .58  makes  2.795,  the  median. 

Coming  down,  we  find  that  we  have  68  cases  from  3  up.     To  get 

100}  cases,  we  must  take  32}  cases  from  the  upper  end  of  group  2. 

Figuring  as  before,  °-§£-  of    93  =  .35.     3.145  -  .35  gives  us  2.795 

86 
for  the  median,  the  same  result  as  before. 

Some  books  give  rules  for  finding  the  median  which 
involve  finding  the  middle  case.  The  middle  case  does 
represent  the  median  in  that  its  magnitude  is  the  median. 
But  there  is  danger  in  using  the  formula  of  adding  1  to 
the  total  number  of  cases  and  dividing  by  2  to  get  the 
middle  case.  The  danger  lies  in  the  tendency  to  add  the 
whole  of  the  middle  case  to  the  part  taken  from  the 
group.  Thus  in  the  example  preceding,  a  beginner  is  apt 
to  take  f|  of  step  2  to  add  to  the  lower  limit  of  that 
step,  or  to  take  ff  from  the  upper  limit  if  he  is  coming 
down.  The  former  procedure  really  puts  101  cases  below 
and  only  100  cases  above  the  median  calculated.  The 
other  procedure  puts  101  cases  above  and  only  100  cases  be- 
low the  median  calculated.  Obviously,  two  different  re- 
sults would  come  from  these  two  calculations.  That  is. 
the  median  calculated  coming  down  would  not  agree  with 
the  median  calculated  coming  up.  The.  mid-point  plan, 
however,  will  give  the  same  result  from  either  end  and 
is  consequently  safer. 

In  order  to  emphasize  the  fact  that  the  median  is  best 
located   by   taking   the    mid-point   on    the    scale   or   dis- 


134  School  Statistics  and  Publicity 

tribution,  irrespective  of  whether  the  number  of  cases  is 
odd  or  even,  let  us  take  several  other  examples. 

In  1916,  the  Courtis  Tests  were  given  in  certain  schools  in  a  western 
city,  with  the  following  results  for  the  number  of  problems  attempted 
by  each  eighth  grade  child  in  one  process : 

No.  problems  attempted        No.  attempting 


24 

5 

23 

2 

(7)  Adding  down 

22 

2 

(9) 

21 

1 

(10) 

20 

5 

(15) 

19 

5 

(20) 

18 

5 

(25) 

17 

7 

(32) 

16 

13 

(45) 

15 

15 

(60) 

14 

10 

(70) 

13 

15 

(85) 

12 

24 

(109) 

11 

22 

10 

26 

(88) 

9 

21 

(62) 

8 

19 

(41) 

7 

9 

(22) 

6 

8 

(13) 

5 

3 

(5) 

4 

1 

(2)  Adding  up 

3 

1 

2)219 

+* 

109.5 

There  are  219  cases.  Therefore,  the  median  will  fall  in  the  middle  of 
the  110th  case,  since  there  must  be  109!  cases  on  either  side  of  it. 
Steps  ■!  to  10  inclusive  take  in  88  cases.  Subtracting  88  from  109.5, 
we  find  that  \v<>  must  have  21.5  cases  out  of  the  next  step.  There  are 
2L'  cases  in  the  step.  Therefore,  —j—  of  the  distance  of  this  step, 
which  is  1,  is  .98.     By  the  nature  of  the  Courtis  Tests,  however,  step 


Measures  of  Type  135 

11  extends  from  11  to  12.  Therefore,  the  median  is  11  +  .98 
=  11.98.     Figuring  down,  we  would  have  —  of  1  =  .02  to  be  taken 

from  the  upper  limit  of  step  11,  which  is  12.  12  —  .02  =  11.98  for 
our  median,  the  same  result  as  before. 

Suppose,  now,  that  we  manipulate  the  data  in  the  above  problem 
so  that  step  11  contains  only  1  case  instead  of  22,  the  remaining  21 
being  given  to  step  10,  making  its  total  47.  The  total  of  all  the 
cases  is  unchanged.  We  find,  however,  that  steps  3  to  10  inclusive 
now  contain  109  cases,  and  the  median  must  fall  in  the  11th  step, 
which  has  only  one  case.  It  will,  therefore,  fall  in  the  middle  of  the 
step,  since  the  one  case  must  be  considered  as  extending  over  the  whole 
space  in  the  step.  The  median,  therefore,  in  this  supposed  case  would 
be  11.5. 

In  order  to  show  the  working  of  the  same  principle  in  the  case  of 
the  fifth  grade  composition  results  at  Butte,  given  above,  let  us  ma- 
nipulate the  data  until  they  appear  as  follows : 


Rated  at 

No. 

Papers 

5 

23 

4 

28       (51)  Adding  down 

3 

49     (100) 

2 

1 

1 

46     (100)  Adding  up 

0 

54 

2)201 
100.5 

The  total  number  of  cases  remains  the  same,  and  the  median  would 
fall  as  above,  namely,  at  the  middle  of  the  101st  case,  or  at  the  100 \ 
case.  Steps  0  and  1  now  together  contain  100  cases.  Therefore,  \ 
case  must  be  taken  from  the  next  step.  As  there  is  only  one  case  in 
this  step,  the  median  will  fall  at  the  mid-point  of  this  step,  for  the  same 
reason  as  stated  in  the  solution  of  the  problem  just  above  this  one. 
The  distance  covered  by  step  2  on  the  Hillegas  scale,  as  shown  before, 
is  .93.  One-half  of  this  is  .465.  Add  .465  to  2.215  (the  lower  limit 
of  the  step)  and  we  have  for  the  median,  2.68.    - 

Suppose,  however,  that  we  have  an  even  number  of  cases,  and  the 
median  falls  in  a  step  containing  0  cases,  that  is.  in  a  gap  in  the  dis- 
tribution. For  example,  let  the  data  used  in  the  last  problem  be  ma- 
nipulated so  as  to  appear  thus : 


136  School  Statistics  and  Publicity 


ed  at 

No.  papers 

5 

23 

4 

28 

(51)  Adding  down 

3 

49 

(100) 

2 

0 

1 

46 

(100)  Adding  up 

0 

54 

2)200 

100 

In  this  example,  the  median  would  come  at  the  end  of  the  100th  case. 
We  find  that  the  end  of  this  case  coincides  with  the  end  of  step  1. 
But  coming  down,  the  end  of  the  100th  case  would  be  at  the  begin- 
ning of  step  3.  Obviously  any  point  in  step  2  could  be  taken  and 
there  would  still  be  100  cases  on  each  side.  In  a  problem  of  this 
kind,  the  best  procedure  is  to  divide  the  vacant  space  on  the  scale 
so  that  the  two  nearest  actual  cases  on  it  will  be  equidistant.  Then 
the  median  will  fall  in  the  middle  of  step  2  or  at  2.68  for  the  real  value, 
and  will  be  the  same  as  if  one  case  had  been  in  the  step  with  the  total 
number  of  cases  odd. 

If  the  data  given  on  page  134  from  the  tests  in  arithmetic  are  ma- 
nipulated so  that  there  are  only  218  cases  in  all  and  one-half  of  these 
are  above  11  and  one-half  below,  with  none  in  step  11,  but  actual 
cases  in  steps  10  to  12,  precisely  the  same  procedure  would  be  followed. 
That  is,  in  this  case,  the  median  would  be  11.5. 

Sometimes  the  median  is  not  calculated  exactly  but 
only  approximately.  Professor  Monroe,  for  example, 
gives  some  devices  for  obtaining  an  approximate  median 
and  then  correcting  it  to  get  the  true  median.1  But  in 
general,  about  the  only  time  that  an  apprc?ximate  median 
may  be  used  to  advantage  is  when  one  takes  for  it  the 
magnitude  of  the  lower  limit  of  the  group  that  contains 
the  real  median.  That  is,  11  would  be  taken  for  the 
approximate  median  for  the  problem  on  page  134,  since 
the  median  falis  within  the  11-11.99  group.     The  straight 

'  Monroe,  W.  S. :  Educational  Tents  and  Measurements,  pp.  242- 
247 


Measures  of  Type 


137 


mid-point  method  of  calculating  the  median  will  get  it 
with  less  confusion,  greater  accuracy,  and  more  speed  for 
the  average  person. 

Graphic  Representation  of  the  Median.  In  a  Bobbitt 
table  having  an  odd  number  of  items,  the  median  is  the 

No.  attempting 


i 

l 

1 

25 

- 

- 

20 

1 

1 

l 

| 

I 
• 

i 

t 
I 

I 

- 

13 

~~t 

! 

1 
1 

1 
1 

!    I 

!  t 

- 

1 

tj 

I 

1 
1 

I 

— 

__ 

— 

! 

t 

;      ' 

1 

J 

• 

i 

No.   Pr 

3 

>bl< 

;m 

3 

1 

> 

G 

9. 
uar 

lO    , 

M  e< 
tile 

i.' 
lie 

I 

)8 
n 

Qi 

— 

1  5 

15. 

H 

35 

ile 

3 

i 

?. 

5 

Fig.  -20.  —  Graphic-  Representation  of  t  !.<■  Me 
The  uorruapondini;  distribution  tabic 


if  I  re(iueney. 


,,:«.■  I:;-1. 


138  School  Statistics  and  Publicity 

size  of  the  middle  item,  and  it  is  usually  inclosed  in  parallel 
horizontal  lines  to  indicate  that  it  is  the  median.  (See 
page  18.)  If  the  table  has  an  even  number  of  items,  a 
single  horizontal  line  is  drawn  across  between  the  two 
middle  cases,  thus  throwing  half  the  cases  above  and  half 
below.  (See  page  130.)  The  line  then  represents  the 
median,  but  the  exact  magnitude  is  not  shown  in 
figures. 

To  represent  the  median  graphically  on  a  surface  of 
frequency,  find  the  point  on  the  base  line  of  the  surface 
of  frequency  represented  by  the  calculated  median  and 
through  it  erect  a  perpendicular.  This  perpendicular  is 
sometimes  erroneously  called  the  median,  but  really  the 
median  is  the  size  of  the  magnitude  on  the  horizontal 
scale  at  the  foot  of  the  perpendicular.  (See  Figure  20.) 
This  perpendicular  divides  the  surface  of  frequency  into 
two  equal  parts. 

Through  the  point  on  the  base  representing  the  median  (11.98) 
draw  a  perpendicular  (represented  by  a  dotted  line  here)  which  will 
cut  the  surface  into  two  equal  areas.  The  small  squares  may  be 
counted  to  prove  this.  There  are  88  small  squares  to  the  left  of  the 
11-11.99  group  and  109  to  the  right  of  it.  The  11-11.99  group 
contains  22  small  squares,  of  which  .98  must  go  with  the  ones  to  the 
left,  and  .02  with  the  ones  to  the  right.  .98  of  22  =  21.56.  Adding 
to  88  we  get  109.56.  .02  of  22  =  .44.  Adding  to  109,  we  get  109.44, 
or  approximately  the  same  area.  The  slight  discrepancy  is  due  to 
carrying  the  decimals  out  to  only  two  places. 

Advantages  of  the  Median  for  School  Work.     1.   The 

median  can  usually  be  located  exactly  without  much 
trouble. 

This  is  of  great  service  where  the  mode  cannot  be  exactly  deter- 
mined, as  in  distributions  where  practically  any  legitimate  groupings 
will  give  several  groups  of  about  the  same  size. 


Measures  of  Type  139 

2.  Extreme  cases  influence  it  little.  In  this  it  resembles 
the  mode. 

For  example,  dropping  off  a  number  of  cases  at  either  end  of  the 
distribution  table  on  page  134  would  only  shift  the  median  a  part  of 
one  step.  As  long  as  the  median  stays  within  the  22  cases  of  step  1 1 , 
for  instance,  one  case  dropped  at  either  end  could  shift  the  median 
only  2V  or  .045  of  one  step. 

3.  Its  location  can  never  depend  upon  a  small  number 
of  items. 

It  falls  at  the  midpoint  of  the  distribution  irrespective  of  how 
many  cases  are  in  a  group  or  where  the  groups  are. 

4.  If  the  number  of  extreme  cases  is  known,  or  known 
approximately,  we  do  not  have  to  know  their  size. 

Thus,  if  we  wish  to  know  the  median  or  typical  salary  of  Latin 
teachers  in  a  state,  it  is  not  necessary  to  get  the  salaries  of  teachers  in 
the  large  high  schools  that  will  not  report  to  the  state  superintendent, 
nor  in  the  small  unorganized  high  schools,  provided  we  know  about 
how  many  teachers  are  in  each  class  of  schools.  If  we  know  these 
numbers  approximately,  we  can  still  get  the  median  salary  of  Latin 
teachers  for  the  whole  state. 

5.  The  median  is  of  special  value  for  data  where  the 
items  cannot  be  measured  in  definite  units. 

Thus,  we  may  get  the  median  child  on  any  particular  ability  for  a 
room,  or  the  median  performer  in  a  debating  or  oratorical  contest, 
without  ever  being  able  to  measure  in  definite  units  the  performance 
of  a  single  contestant.  An  arithmetical  average  cannot  be  calculated 
here  with  any  useful  accuracy,  but  the  median  can  be  found  and  com- 
pared with  similar  medians. 

Disadvantages  of  the  Median  for  School  Statistics. 
1.    The  median  is  not  so  easily  calculated  as  the  average. 

f 
The  average  is  computed  by  an  arithmetical  process  familiar  to 

most  children  in  the  grammar  grades,  and  it  may  be  computed  without 


140  School  Statistics  and  Publicity 

rearranging  the  items.  The  median  cannot  be  calculated  until  the 
data  are  rearranged  in  order  of  size,  and  while  the  process  of  calcula- 
tion is  simple,  the  previous  examples  show  that  considerable  care 
must  be  exercised  in  getting  it. 

2.  The  total  cannot  be  gotten  by  multiplying  the 
median  by  the  number  of  items.  In  this  respect  it  is 
like  the  mode. 

3.  It  is  not  useful  in  those  cases  where  it  is  desirable 
to  give  large  weight  to  extreme  variations. 

Thus,  the  average  daily  attendance  of  a  school  is  much  affected  by 
those  who  are  absent  a  large  part  of  the  time.  The  median  attendance 
would  not  be  so  affected.  But  what  we  wish  in  this  particular  in- 
stance is  that  the  great  effect  of  the  few  extreme  cases  shall  exert  its 
full  influence.     Consequently,  the  average  daily  attendance  is  used. 

4.  Unlike  the  mode  but  like  the  average,  the  median 
may  be  located  in  the  distribution  where  the  actual  cases 
are  few. 

5.  In  a  discrete  series  there  may  be  so  many  cases  the 
same  size  as  the  median  that  it  will  become  almost 
meaningless. 

It  cannot  mean  much  here  unless  there  is  some  reasonable  basis 
for  regarding  the  given  measures  as  spread  out,  that  is,  for  regarding 
the  distribution  as  in  some  ways  continuous. 

EXERCISES 

1.  Calculate  the  median  from  both  ends  for  each  of  the  distribu- 
tion tables  used  in  previous  exercises. 

2.  Draw  the  line  to  represent  the  position  of  the  median  in  each 
of  the  surfaces  of  frequency  used  in  previous  exercises. 


Measures  of  Type  141 

III.    THE   AVERAGE 

Definition.  The  average  is  a  measure  much  used  in 
ordinary  life  without  being  denned.  Indeed,  it  is  not 
capable  of  easy  definition.  The  ordinary  person,  if 
pinned  down  long  enough,  will  define  it  as  a  measure 
which  will  give  "  the  general  run  "  of  a  group  by  taking 
into  account  both  the  number  of  cases  and  the  size  of 
each  one. 

But  actually  he  means  one  of  two  things,  which  differ 
widely  from  each  other  and  neither  of  which  really 
corresponds  to  this  definition.  Most  of  the  time  the 
"  average  "  means  to  him  the  most  frequent  measure  in 
the  group,  i.e.,  the  mode.  But  in  some  instances,  it 
means  to  him  a  very  unreal  and  unjust  thing  obtained 
by  statistical  sleight  of  hand.  Thus  he  will  say  that 
there  is  no  such  thing  as  an  average  boy,  nobody  ever 
saw  such  a  boy,  etc.  This  comes  nearer  his  definition 
than  does  his  other  use  of  the  term.  But  even  here  he 
probably  is  not  conscious  of  the  fact  that  the  average 
is  really  the  size  of  the  balancing  point  or  center  of 
gravity  in  the  distribution. 

Calculation.  1.  Ordinary  method.  Ordinarily,  the 
average  is  calculated  by  dividing  the  sum  of  all  (he 
measures  (or  cases)  by  the  number  of  measures.  By 
formula  it  is  : 

Sum  of  all  measures 
Average  (Av.)  =       VT        .—  — — 

H  No.  <>l  measures 

Thus,  for  the  results  on  the  Courtis  Tests,  page  I'M.  we  could  find 
the  average  as  follows : 


142 


School  Statistics  and  Publicity 


No: 

Problems 

No. 

All 

Attempted 

A  Uempting 

Measures 

24 

X 

5 

120 

23 

X 

2 

46 

22 

X 

2 

44 

21 

X 

1 

21 

20 

X 

5 

100 

19 

X 

5 

95 

18 

X 

5 

90 

17 

X 

7 

119 

16 

X 

13 

208 

15 

X 

15 

225 

14 

X 

10 

140 

13 

X 

15 

195 

12 

X 

24 

288 

11 

X 

22 

242 

10 

X 

26 

260 

9 

X 

21 

189 

8 

X 

19 

152 

7 

X 

9 

63 

6 

X 

8 

48 

5 

X 

3 

15 

4 

X 

1 

4 

3 

X 

1 

3 

219 

219)2667 

12.18  Average  No. 
Problems  Attempted. 

2.  Short  method.  By  experienced  workers  in  statistics 
averages  are  often  computed  by  the  following  method  of 
guessing  the  average  and  then  correcting  it. 

First  arrange  figures  in  a  distribution  table.  Then 
guess  the  average  by  inspection,  usually  taking  the 
approximate  median.  To  be  absolutely  correct,  this 
must  be  guessed  within  one  step  of  the  true  average. 

Then  correct  the  guessed  average  by  the  average  of  all 
the  deviations  from  it. 


Measures  of  Type 


143 


In  the  Courtis  Test  problem  just  worked,  the  procedure  is  as  fol- 
lows: 

Guessed  average  is  11. 
Deviations  above  or  +  deviations  are : 


+ 
24  deviations  of       1 

15 2 

10 3 

15 4 

13 5 

7 6 

5 7 

5 8 

5 9 

1 10 

2 11 

2 12 

5 13 

Deviations  below  or  —  deviations  are  : 


+ 
24 
30 
30 
60 
65 
42 
35 
40 
45 
10 
22 
24 
65 
-492 


26  dev 

iations  o 

1      or 

21   . 

•2 

19  . 

3     .     . 

9  . 

4     .     . 

8  . 

5     .     . 

3   . 

6     .     . 

1   . 

7     .     . 

1   .     . 

8     .     . 

26 

42 
57 
36 
40 
18 
7 
_S 
-234 
+  Deviations  492 
234 
Excess  of  +  deviations    =  258 

This  means  that  we  got  the  guessed  average  too  low  because  there 
are  more  deviations  above  than  below  it.  The  258  is  the  excess  of 
deviations  above.  As  there  are  219  cases,  the  average  excess  devia- 
tion to  satisfy  every  case  is  l\\  or  LIS  of  a  step.     As  the  guessed  aver- 


144  School  Statistics  and  Publicity 

age  of  11  was  too  low,  we  add  this  1.18  average  deviation  and  get 
12.18  for  our  corrected  average. 

Had  the  minus  deviations  been  in  excess,  it  would  have  meant 
that  the  guessed  average  was  too  high,  and  we  should  have  sub- 
tracted the  correction. 

The  rule  for  computing  the  average  by  the  short-cut  method 
is:  "  Arrange  the  numbers  in  the  order  of  their  magni- 
tude ;  choose  any  number  likely  to  be  nearest  the  average  ; 
add  together,  regarding  signs,  the  deviations  from  it  of  all 
the  numbers ;  divide  this  result  by  the  number  of  the 
measures  of  the  average  which  you  are  obtaining;  add 
the  quotient  to  the  chosen  number."  x 

In  using  the  short-cut  method,  three  cautions  must  be 
kept  in  mind : 

1.  To  get  much  accuracy,  the  guessed  average  must  be  within  one 
step  of  the  true  average, 

2.  The  correction  must  be  added  to  the  guessed  average  if  the 
plus  deviations  are  in  the  majority;  it  must  be  subtracted  if  the 
minus  deviations  are  in  the  majority. 

3.  If  the  guessed  average  is  in  a  group,  the  deviations  for  all  that 
group  are  0.  The  number  of  cases,  however,  must  be  used  in  dividing 
to  get  the  average  deviation. 

The  short  method  is  not  always  short  by  itself ;  but  it 
often  saves  time  on  certain  calculations  on  deviations. 

Graphic  Representation.  The  average  can  be  rep- 
resented graphically  in  the  same  way  "*»s  the  median. 
(See  page  137.)  That  is,  the  point  representing  the 
calculated  average  is  found  on  the  base  line  and  a  per- 
pendicular is  erected  to  call  attention  to  the  size  of  the 
average.  However,  there  is  in  general  little  value  in 
representing  the  average  this  way  because  the  line  drawn 
has  no  appreciable  relation  to  the  surface  of  frequency. 

1  Thorndike,  E.  L. :   Mental  and  Social  Measurements,  p.  3 


Measures  of  Type  145 

A  line  for  the  mode  runs  through  the  highest  part  of 
the  surface,  a  fact  the  eye  easily  grasps.  A  line  for  the 
median  cuts  the  surface  into  two  equal  areas,  which  the 
eye  will  readily  compare.  But  the  average  has  no  such 
relationship  to  show. 

Advantages  of  the  Average  for  School  Statistics. 
1.  "  Unlike  the  median  or  mode,  it  may  be  definitely 
located  by  a  simple  process  of  addition  and  division,  and 
it  is  unnecessary  to  draw  diagrams  or  arrange  the  data  in 
any  set  form  or  series."  x 

The  great  ease  with  which  the  average  can  be  calculated  from 
figures  in  almost  any  form  is  doubtless  the  main  reason  it  has  been  so 
often  used.  It  is  not  even  necessary  to  throw  figures  into  a  distribu- 
tion table  or  table  of  frequency,  as  is  the  case  in  finding  the  median 
or  the  mode. 

2.  It  weighs  extreme  cases,  which  is  a  desirable  thing 
in  certain  instances. 

3.  "  Unlike  the  mode,  it  is  affected  by  every  item  in  the 
group,  and  its  location  can  never  be  due  to  a  small  class 
of  items."  2 

Thus  Superintendent  Spaulding  in  a  recent  bulletin  gets  the  aver- 
age expenditure  for  Minneapolis  by  taking  the  average  for  five  years. 
No  one  year  is  any  more  important  than  another  and  so  ^ets  counted 
no  more  and  no  less  than  another  year.  The  median  in  such  a  small 
number  of  cases,  with  an  odd  number  especially,  would  emphasize 
one  year  unduly. 

4.  The  method  of  calculating  it  is  familiar  to  every  one. 

5.  It  may  be  determined  when  the  aggregate  and  the 
number  of  items  are  the  only  things  known. 

Thus,  in  determining  the  typical  number  of  days'  attendance  for 
a  child  in  a  certain  grade,  we  may  have  the  total  number  of  days' 

fl  King,  W.  I. :   Element*  of  Statistical  Method,  p.  136 
'■Ibid.,  p.  136 


146  School  Statistics  and  Publicity 

attendance  and  the  total  number  of  children.  We  might  not  have  the 
actual  record  of  a  single  child  given  us.  But  from  the  other  two  items 
we  could  calculate  the  average  number  of  days'  attendance  for  a  child. 
Sometimes  these  are  the  only  two  items  that  can  be  secured  from 
reports.  Thus  we  may  have  only  the  total  for  teachers'  salaries  and 
the  number  of  teachers ;  the  total  sum  spent  on  repairs  and  the  num- 
ber of  buildings;  the  total  expenditures  for  a  certain  kind  of  school 
and  the  number  of  such  schools,  etc. 

Disadvantages   of  the   Average   for   School   Statistics. 

1.  It  cannot  be  located  on  a  surface  of  frequency  from 
looking  at  the  surface  alone. 

The  median  or  mode  can  be  quickly  located  by  inspecting  the  sur- 
face of  frequency.  The  average  can  be  put  on  it  only  after  the  aver- 
age has  been  calculated,  and  the  point  counted  out  on  the  base  scale. 

2.  It  cannot  be  located  accurately  if  the  extremes  are 
missing  or  in  any  way  doubtful. 

This  is  particularly  troublesome  when  we  recall  that  we  are  often 
inclined  to  expect  extreme  cases  of  expenditures  and  such  things 
in  school  work  to  be  doubtful  or  inaccurate. 

3.  It  lays  too  much  stress  on  extreme  variations. 

This  is  as  bad  a  thing  for  school  statistics  as  judging  a  patent 
medicine  by  the  enthusiastic  few  who  write  testimonials  about  it  for 
publication,  or  a  school  by  what  is  done  the  first  and  last  week  in  each 
session.  The  trouble  can,  however,  be  eliminated  to  some  extent 
by  dropping  very  extreme  cases  and  calculating  the  average  from  the 
remaining  ones. 

4.  It  cannot  be  used  where  we  cannot  accurately 
measure  the  quantities  studied. 

The  so-called  averaging  of  points  on  contestants  in  a  debate  is  often 
futile  because  they  are  not  measured  by  the  different  judges  on  the 
same  scale. 

5.  It  may  fall  where  no  data  actually  exist. 

That  is,  it  may  fall  in  a  gap  (the  median  does  this  sometimes  as 
well)  and  be  almost  as  absurd  as  the  case  of  the  duck  reported  by  a 


Measures  of  Type  147 

western  professor.  The  duck  was  shot  at  with  a  double-barreled 
shotgun,  one  shot  going  two  feet  to  the  right  of  the  duck  and  the  other 
two  feet  to  the  left.  The  average  performance  was,  of  course,  zero 
and  centered  on  the  duck.  Statistically  the  duck  was  dead ;  actually 
it  flew  away. 

6.  The  average  often  means  to  the  ordinary  man 
something  different  from  what  the  calculated  thing  means. 

The  process  of  calculation  is  undoubtedly  familiar  to  all.  But 
what  the  ordinary  man  often  means  by  the  "average  boy"  is  a  boy 
like  the  majority  of  boys,  not  like  the  few  above  or  below  this  majority. 
The  man  in  all  probability  has  seldom  thought  through  the  fact  that 
his  method  of  calculation  does  not  always  give  him  this  "majority" 
measure. 

EXERCISES 

1.  Calculate  by  both  the  long  and  short  method  the  average  for 
each  of  the  distribution  tables  used  in  previous  exercises. 

2.  Draw  the  line  to  represent  the  position  of  the  average  in  each 
of  the  surfaces  of  frequency  used  in  previous  exercises. 

IV.    WHICH  MEASURE  OF  TYPE  TO  USE  IN  A  GIVEN 
DISTRIBUTION 

If  the  distribution  is  symmetrical  or  approximately  so, 
the  average,  median,  or  mode  are,  of  course,  the  same  or 
approximately  so.  Consequently,  as  regards  magnitude, 
it  is  a  matter  of  indifference  which  of  the  three  is  used  as 
a  measure  of  central  tendency.  But  it  is  generally  im- 
possible to  tell  whether  the  distribution  is  symmetrical 
or  not  till  a  frequency  table  has  been  made.  Once  this 
table  has  been  made,  it  is  much  easier  to  determine  the 
median  than  the  average.  In  a  Bobbitt' table  or  similar 
distribution  where  all  cases  are  separate,  the  median 
can  be  found  much  more  quickly  than  the  average. 

The  mode  is  the  measure  to  use  in  all  skew  distributions 
or  bi-modal  ones, 


148  School  Statistics  and  Publicity 

-The  matter  of  symmetry  or  skewness  can  be  seen  at  a 
glance  from  the  surface  of  frequency.  With  a  little 
experience  it  can  be  recognized  directly  from  the  table  of 
frequency. 

For  school  statistics,  this  is  probably  the  safest  rule : 
Use  the  mode  for  skew  and  bi-modal  distributions; 
the  average,  for  cases  where  every  item  must  be  counted 
as  much  as  any  oilier  item  (say  in  finding  the  average 
expenditures  for  five  or  less  years) ;  and  the  median  for 
all  others. 

Of  the  three,  the  median  is  by  far  the  best  "  all  pur- 
pose "  measure  of  central  tendency  for  school  statistics. 

EXERCISE 

In  each  of  the  distribution  tables  for  which  you  have  calculated 
the  mode,  median,  and  average,  which  is  the  best  measure  of  central 
tendency?      Why? 

REFERENCES    FOR   SUPPLEMENTARY    READING 

King,  W.  I.     Element*  of  Statistical  Method,  Chapter  XII. 

Rugg,  H.  O.     Statistical  Methods  Applied  to  Education,  Chapter  V. 

Thorndike,  E.  L.     Mental  and  Social  Measurements,  pp.  36-39. 


CHAPTER   VI 
MEASURES    OF   DEVIATION    OR   DISPERSION 

The  second  element  necessary  in  giving  the  bird's-eye 
view,  it  will  be  recalled,  is  some  measure  of  how  much 
the  cases  range  from  the  central  tendency.  This  is 
variously  called  "  spread,"  "  range,"  "  deviation,"  or 
"  dispersion."  It  is  of  just  as  much  importance  as  the 
central  tendency.  For  example,  it  makes  a  vast  difference 
whether  a  teacher  is  called  upon  to  teach  children  that 
are  all  very  close  to  the  typical  or  "  average  "  child,  or 
ones  that  vary  enormously  both  above  and  below. 
Supervision  means  one  thing  for  a  superintendent  when 
all  his  teachers  are  very  close  in  ability,  experience, 
spirit,  etc.,  to  his  typical  teacher;  it  means  an  entirely 
different  thing  when  many  of  his  teachers  range  widely 
either  side  of  this  typical  teacher. 

There  are  several  ways  of  measuring  this  range,  the 
most  useful  of  which  will  now  be  given. 

I.    EXTREME   RANGE    VARIATION 

The  dispersion  may  be  shown  by  giving  the  extreme 
cases  and  thus  showing  how  far  it  is  between  them.  This 
measure  may  be  useful  in  showing  the  difference  of  ability 
in  achievements  of  a  grade  in  school.  • 

For  example,  the  eighth  grade  in  a  western  city  as  reported  on 
some' Courtis  Tests  varied  in  attempts  on  the  addition  problems 
from  4  to  24,  or  a  range  of  20  out  of  a  possible  range  of  24.     This 

149 


150  School  Statistics  and  Publicity 

measure  shows  at  once  that  there  is  a  wide  range  of  ability  of  eighth 
grade  children  in  that  city  as  regards  attempts  on  addition  problems. 

But  as  a  general  rule,  the  extreme  range  variation  is  not 
a  good  measure  of  dispersion  because  the  extreme  cases 
are  likely  to  be  very  unreliable,  especially  if  there  are 
only  one  or  two  that  are  isolated  from  the  rest  of  the 
measures  in  the  distribution. 

The  range  of  teachers'  salaries  in  the  typical  school  system  is 
not  from  that  of  the  superintendent  down  to  the  salary  of  the  poorest 
paid  teacher,  because  there  lb  only  one  superintendent  and  his  salary 
is  far  removed  on  the  scale  from  that  of  the  highest  paid  teacher  under 
him.  His  salary  must  be  eliminated  before  the  extreme  range  for 
the  others  has  any  value. 

The  influence  of  extreme  variations  upon  this  measure  is  shown 
by  Table  17,  taken  from  a  study  in  costs  of  instruction  in  home 
economics  in  fifteen  southern  normal  schools.1 

Table  17.  Variations  in  Cost  of  Instruction  in  Home 
Economics  in  15  Southern  Normal  Schools,  per  1000  Stu- 
dent Hours,  1916 

Cost  of  home  economics 
Key  number  of  instruction  per  1000 

normal  school  student  hours 

III  $194 
VII  115 

XIII  89 
II  88 

VIII  85 

IV  82 
XI                                                            79 

XIV  73 
XV  70 

I  64 

VI  60 

X  58 

V  53 

IX  52 

XII  15 

1  Alexander,  Carter:  "Costs  of  Instruction  in  Normal  Schools," 
Elementary  School  Journal,  XVII,  653 


Measures  of  Deviation  or  Dispersion     151 

The  extreme  range  variation  between  194  and  15  is  179.  But  it 
is  very  evident  that  both  extremes  are  the  results  of  some  very  unusual 
factor.  If  the  top  case  is  cut  off,  the  range  is  reduced  to  100.  If  the 
bottom  case  is  cut  off,  but  the  top  one  retained,  the  range  is  142. 
If  both  are  cut  off,  the  range  is  reduced  to  63.  If  the  two  top  cases 
and  the  bottom  one,  all  of  which  seem  unusual,  are  eliminated,  the 
range  is  reduced  to  37,  which  is  probably  a  fairly  reliable  figure. 

The  uncertainty  as  to  just  how  many  cases  to  leave 
out  in  any  one  distribution  when  computing  the  spread 
emphasizes  the  need  of  having  some  standard  proportion 
cut  off,  say  a  fourth  at  each  end.  Then  the  average 
deviation  or  spread  from  the  central  tendency  of  the  two 
intersections  or  points  thus  obtained  can  be  computed. 
Let  us  now  discuss  some  of  the  devices  for  getting  devia- 
tions from  such  points. 

II.    QUARTILE    DEVIATION    (SEMI-INTER-QUARTILE    RANGE 

OR    "  Q  ") 

This  is  by  far  the  easiest  and  most  important  measure 
of  dispersion  for  school  statistics.  But  before  it  can  be 
easily  understood,  it  is  necessary  to  get  the  quartiles. 
These  are  the  magnitudes  of  the  points  on  the  scale  which 
divide  the  distribution  into  four  equal  groups  of  cases  (or 
quarters) . 

Quartiles.  Obviously,  it  takes  three  such  points  to 
divide  the  whole  into  four  groups  with  equal  numbers  of 
cases,  —  the  first  quartile  or  25  percentile,  the  second 
quartile  (which,  of  course,  is  the  median),  and  the  third 
quartile  or  75  percentile.  Obviously,  also,  the  first 
quartile  is  the  median  of  the  lower  half  of  the  distribution, 
and  the  third  quartile  is  the  median  of  the  upper  half. 
Below  the  first  quartile  come  one-fourth  of  the  cases 
and  above  it,   three-fourths.     Above  the  third   quartile 


152  School  Statistics  and  Publicity 

come  one-fourth  of  the  cases,  and  below  it,  three-fourths. 
Between  the  first  and  third  quartiles  come  one-half  or 
50  per  cent  of  the  cases. 

The  problem  of  locating  the  first  quartile  is  statistically 
the  same  as  that  of  locating  the  median,  except  that  cases 
are  counted  in  from  one  end  so  as  to  get  only  one-fourth 
of  them  on  one  side.  That  is,  the  object  this  time  is  to 
find  the  "  quarter  point  "  from  one  end  of  the  distribution. 

In  a  Bobbitt  table  it  is  generally  best  to  fix  the  number  of  cases  so 
as  to  know  in  advance  whether  the  quartiles  will  be  shown  as  separate 
cases  or  fall  between  separate  cases.  In  this  way,  there  will  be  no 
trouble  with  splitting  cases.  Thus  if  the  number  of  cases  is  some 
multiple  of  4,  plus  1,  the  median  will  appear  as  a  separate  case  and 
the  quartiles  will  fall  between  cases.  For  the  Bobbitt  table  on  page 
18,  the  quartiles  are  respectively  47  for  Q  1  (i  of  42  +  52)  and  78  for 
Q  3  (2  of  74  +  82).  .  If  the  total  number  of  cases  is  some  multiple  of 
4,  the  median  and  both  quartiles  will  fall  between  cases.  If  the  total 
number  of  cases  is  some  multiple  of  4,  plus  3,  the  median  and  both 
quartiles  will  appear  as  separate  cases. 

The  calculation  of  the  quartiles  in  the  distribution  given  on  page 
134  is  as  follows.  Since  there  are  219  cases,  the  first  quartile  must 
come  at  the  point  that  will  throw  \  of  219  or  54  ]  cases  below  it. 
Counting  up,  we  find  that  this  will  require  13. *  cases  of  the  9-10  group, 

13  75 
there  being  only  41  cases  below.     This  will  make  it  come  — ^ — ■  or 

21 
.65  of  the  step  up.     Q  1  then  is  9  +  .65  =  9.65.     Similarly  Q  3  is 

9  75 
in     the     15    group     coming     down.      54J  —  45  =  9|.     -L—  =    .65. 

Q  3  then  is  16  -  .65  =  15.35.  *r 

Graphically,  the  first  quartile  is  the  magnitude  on  the 
base  at  the  foot  of  a  perpendicular  so  located  that  it  will 
cut  off  one-fourth  of  the  area  of  the  surface  of  frequency 
to  the  left  of  it  and  three-fourths  to  the  right.  The  third 
quartile  lias  one-fourth  of  the  area  to  the  right  and  three- 
fourths  to  the  left.  The  median  has  one-half  the  area 
on  either  side  of  it.     (See  Figure  20,  page  137.) 


Measures  of  Deviation  or  Dispersion     153 

Quartile  Deviation.  The  Q  is  found  by  taking  half 
the  difference  between  the  first  and  third  quartiles.  This 
average  will  give  the  average  distance  of  the  quartiles 
from  the  median  of  central  tendency  selected.  The  aver- 
age distance  is  used  because  sometimes  the  median  is  not 
exactly  halfway  between  the  quartiles  (it  cannot  be  if  the 
distribution  is  not  perfectly  symmetrical). 

By  formula,  then, 

~  _  Quartile  3  —  Quartile  1 
H  -—  — 2~ 

For  the  Bobbitt  table  on  page  18,  the  calculation  is : 

Q  =  $78  "  $47  =  1|*  =  $15.50 

For  the  distribution  on  page  134,   the    calculation  is: 

~       15.35  -  9.65      5.70  _  9  er 
Q  =  2        — —  _^.bo 

Graphically,  Q  is  one-half  the  distance  from  Quartile  3 
to  Quartile  1  on  the  base  of  the  surface  of  frequency.  But 
as  it  is  difficult  to  show  only  one-half  of  this,  the  whole  of 
this  distance  is  usually  represented  as  2  Q.     (See  Figure  20.) 

III.    OTHER   PERCENTILE    DEVIATIONS 

In  some  of  the  educational  investigations,  especially 
those  issued  by  workers  at  the  University  of  Chicago, 
dispersion  is  indicated  by  the  distance  from  the  median 
of  some  point  marking  off  a  convenient  number  of  the 
cases,  other  than  the  quartile.  The  number  of  cases  is 
usually  a  common  percentage;  hence. the  point  is  called 
a  "  percentile."  In  some  studies  the  cases  are  divided 
into  thirds  by  "  tertiles."  The  median  here  is,  of  course, 
in  the  middle  of  the  second  group.  The  fertile  deviation, 
then,  is  half  the  difference  of  the  first  and  second  tertiles 


154 


School  Statistics  and  Publicity 


(only  two  tertiles  are  needed  to  make  three  groups). 
Similarly  four  "  quintiles  "  will  divide  the  distribution 
into  five  groups  with  the  same  number  of  cases,  the  median 
being  at  the  middle  of  the  third  group.  The  quintile 
deviation  then  would  be  one-half  the  difference  of  the 
first  and  fourth  quintiles. 

Table  18.  Table  to  Show  Variation  by  Percentage  Groups, 
Using  Distribution  of  Annual  Salaries  of  Regular  Teach- 
ers in  Elementary  Schools  in  Cleveland  and  in  13  Other 
Cities  of  More  Than  250,000  Inhabitants1 


Salaries  not  exceeding  the  amounts  specified  toere 

earned  by  teachers  bearing  to  the  aggregate  num- 

City 

ber  employed  in  each  city  the  proportion  of: 

10  per 

30  per 

50  per 

70  per 

90  per 

cent 

cent 

cent 

cent 

cent 

Baltimore 

$600 

$    700 

%    700 

$    750 

$    800 

Boston     .... 

648 

840 

1,176 

1,176 

1,224 

Chicago         .     .     . 

675 

975 

1,175 

1,175 

1,200 

Cincinnati    .     . 

700 

900 

1,000 

1,000 

1,000 

Cleveland  .     .     . 

600 

750 

900 

1,000 

1,000 

Indianapolis 

475 

625 

875 

925 

925 

Milwaukee 

876 

876 

876 

876 

876 

Minneapolis 

750 

950 

1,000 

1,000 

1,000 

Newark    .... 

630 

780 

1,000 

1,000 

1,300 

New  Orleans 

500 

600 

700 

^     750 

800 

Philadelphia 

630 

780 

900 

940 

1,000 

San  Francisco   . 

840 

1,164 

1,200 

1,224 

1,224 

St.  Louis 

700 

1,032 

1,032 

1,032 

1,120 

Washington 

625 

700 

$    831 

750 
$    949 

890 
$~~988 

980 

Average     . 

$661 

$1,032 

1  Data  for  Cleveland  from  payroll  for  1914  15;  data  for  other 
cities  for  1913  14,  from  "Tangible  Rewards  of  Teaching,"  U.  S. 
Bureau  of  Education 


Measures  of  Deviation  or  Dispersion     155 

The  calculations  and  graphic  representations  for  the.se 
various  percentile  deviations  are  precisely  similar  to  those 
for  the  Q  or  quartile  deviation. 

These  deviations  are  found  as  yet  in  few  educational 
investigations,  but  should  be  understood  for  reading 
purposes. 

A  simple  but  effective  device  for  popular  consumption 
indicates  the  variability  indirectly  by  showing  the  magni- 
tudes which  equal  or  exceed  certain  fixed  percentages  of 
the  cases.  This  device  has  been  found  especially  valu- 
able for  making  comparisons  between  different  distribu- 
tions. 

A  good  example  occurs  in  the  Cleveland  Survey  l  and 
is  reproduced  in  Table  18. 

IV.    MEDIAN   DEVIATION    (MED.    DEV.   OR  P.    E.) 

This  is  the  median  of  all  the  deviations  arranged  in 
order  of  size  and  irrespective  of  whether  they  are  above  or 
below  the  central  tendency.  In  other  words,  it  is  cal- 
culated by  arranging  the  deviations  in  order  of  size  and 
then  finding  the  median  of  these  deviations  in  precisely 
the  same  way  as  the  median  of  anything  else  would  be 
found. 

Within  the  median  deviation  of  the  central  tendency 
used,  the  middle  50  per  cent  of  the  cases  come.  Within 
the  Q  of  the  central  tendency  approximately  50  per  cent 
of  the  cases  come,  but  they  are  not  necessarily  the  middle 
50  per  cent,  because  the  central  tendency  does  not 
lie  exactly  halfway  between  the  quartiles,  except  in  a 
symmetrical  distribution.  In  such  a  distribution  the 
median  deviation  is  of  course  the  same  as  Q. 

1  Volume  on  "  Financing  the  Schools,"  p.  57 


156  School  Statistics  and  Publicity 

The  median  deviation  is  little  used  in  school  statistics, 
but  the  superintendent  may  come  across  it  in  his  reading 
of  educational  investigations  and  in  attempts  to  indicate 
variability.  It  is  sometimes  called  the  "  probable  error/' 
and  this  is  where  it  gets  the  letters  "P.  E."  However, 
this  name  is  not  a  good  one  for,  as  Professor  Thorndike 
suggests,  the  median  deviation  "  is  not  specially  probable 
and  not  an  error  at  all."  '  A  statement  of  central 
tendency  of  30  with  a  P.  E.  of  4  means  that  half  the  cases 
deviate  more  than  4  and  half  of  them  less  than  4  from  this 
30.  It  also  means  that  for  any  given  case  chosen  at 
random,  the  chances  are  50  to  50  that  it  will  deviate  more 
than  4  from  this  30,  that  is,  be  below  26  or  above  34 ; 
the  chances  are  likewise  50  to  50  that  it  will  deviate  less 
than  4  from  this  30,  that  is,  be  between  26  and  34.  In 
other  words,  it  is  a  toss-up  as  to  whether  any  given  case 
will  be  likely  to  deviate  more  or  less  than  4  from  this 
central  tendency  of  30  ;  that  is,  any  given  case  is  as  likely 
to  be  between  26  and  34  as  it  is  to  be  below  26  or 
above  34. 

V.    AVERAGE   DEVIATION    (A.   D.) 

This  is  simply  the  average  of  all  the  deviations  from 
whatever  central  tendency  is  selected.  It  may  be  figured 
from  any  measure  of  central  tendency.  JBut  it  is  best 
calculated  from  the  average  or  approximate  average, 
next  from  the  median  or  approximate  median,  and  seldom 
if  ever  from  the  mode,  unless  the  average  spread  for  each 
side  is  given  separately. 

The  A.  D.  for  the  Bobbitt  table  on  page  18  is  figured 
thus : 

1  Menial  and  Social  Measurements,  p.  40,  footnote 


Measures  of  Deviation  or  Dispersion     157 

Deviation  from  median  (59)  of  61   is    2 


62 

3 

63 

4 

69 

10 

69 

10 

74 

15 

82 

23 

88 

29 

100 

41 

100 

41 

112 

53 

169 

110 

59 

0 

58 

1 

56 

3 

56 

3 

54 

5 

53 

6 

52 

7 

42 

17 

41 

18 

38 

21 

34 

25 

33 

26 

30 

29 

No.  of  cases,  25)502  sum 

20.1  A.  D. 

The  A.  D.  for  a  distribution  in  groups  is  usually  figured 
from  the  approximate  average  or  the  approximate  median, 
so  as  to  have  the  deviations  in  whole  steps.  The  result 
is  close  enough  for  all  practical  purposes. 

The  A.  D.  for  the  distribution  given  on  page  134  is  figured  thus 
from  the  guessed  average  : 

492  +  deviations  1  ,  .„ 

ooi         ,      •   ..         }  sop  page  143 
234  —  deviations  J         ' 

No.  of  cases,  219)726  sum  of  all  deviations 

'  3.31  A.  D. 


158  School  Statistics  and  Publicity 

VI.  STANDARD  DEVIATION  (MEAN  SQUARE  DEVIATION,  S.  D.) 

> 

This  is  found  by  taking  the  square  root  of  the  average 
of  the  squares  of  the  deviations,  counting  zero  deviations. 

q,  -p.  _  /Sum  of  squares  of  deviations 
*  Number  of  deviations 
It  is  of  no  particular  value  in  school  statistics  except 
that  the  superintendent  shou'd  understand  it  so  that  he 
may  read  intelligently  educational  or  economic  treat- 
ments that  use  it.  It  saves  time  to  use  this  as  the  measure 
of  dispersion  if  the  Pearson  Coefficient  of  Correlation  is 
later  to  be  used.  It  is  better  than  the  A.  D.  if  the  de- 
viations of  the  extreme  cases  need  to  be  weighted. 

The  calculation  of  the  S.  D.  from  the  guessed  average  in  the  dis- 
tribution on  page  134  is  figured  as  follows,  using  the  deviations  as 
given  on  page  143  : 


24  X 

12 

=       24 

15  X 

22 

=       60 

10  X 

32 

=       90 

15  X 

42 

=    240 

13  X 

52 

=    325 

7  X 

62 

=    252 

5  X 

72 

=    245 

5  X 

82 

=     320 

5  X 

.  92 

=     405 

1  X 

102 

=     100 

2  X 

112 

=     242 

2  X 

122 

=     288 

5  X 

132 

=     845 

26  X 

12 

=      26 

21  X 

22 

=       84 

19  X 

•  >2 

-     171 

9  X 

42 

=     144 

8  X 

S2 

=     200 

3  X 

62 

=     108 

1  X 

72 

=      49 

1  X 

82 

=       64 

4282 

S.D.  =  J4282  =4.42 
\  219 


Measures  of  Deviation  or  Dispersion     159 

VH.    DEVIATIONS   FOR    SKEW   DISTRIBUTIONS 

The  measures  for  deviations  so  far  given  are  serviceable 
only  for  symmetrical,  or  approximately  symmetrical, 
distributions.  If  the  distribution  is  a  skew  one,  it  is 
evident  that  no  average  deviation  for  the  whole  can  be 
of  any  service,  because  the  deviation  on  one  side  of  the 
central  tendency  will  be  markedly  different  from  the 
deviation  on  the  other  side.  This  average  or  median 
deviation  for  the  whole  would  be  a  deviation  which  did 
not  actually  exist. 

Hence  in  a  skew  distribution  it  is  customary  to  take 
the  mode  or  median  (preferably  the  mode)  for  the  central 
tendency,  and  then  to  give  the  average  or  median  deviation 
for  the  cases  above  this  central  tendency,  and  the  same 
for  the  cases  below  it. 

This  procedure  may  be  illustrated  from  the  salaries  paid  the 
Cleveland  eleimntary  school  teachers  in  1914-15. '     (See  page  160.) 

The  approximate  median  is  sufficient  here  and  inasmuch  as  only 
an  average  deviation  is  to  be  figured,  it  is  unnecessary  to  make  the 
groupings  by  even  steps. 

There  are  in  all  2204  teachers,  making  1102  above  the  median  and 
1102  below.     The  A.  D.  above  is 


The  A.  D.  below  is 


$1188,10  =$1Q8_ 
1102 

$200920  =  $1S9  , 
1102 


The  median  of  the  part  above  the  group  containing  the  median 
is  at  the  end  of  the  525th  case,  evidently  in  the  $1000  group,  making 
the  quartile  deviation  above  roughly  $100.  The  median  of  the  part 
below  the  group  containing  the  median  is  evidently  in  the  $700 
group,  making  the  quartile  deviation  below  roughly  $200.  The 
exact  median  and  the  exact  quartile  deviation  could  be  figured  if 
desired,  counting  1102  cases  above  the  median  and  551  cases  above 
the  quartile. 

1  Adapted  from  Cleveland  Survey,  Summary  Volume,  p.  98 


160 


School  Statistics  and  Publicity 


Salary  paid 

Number  getting 
salary 

Deviation  from 
median 

Sum  of 
deviations 

$1650 

1 

750 

+  750 

1540 

2 

640 

1280 

1500 

4 

600 

2400 

1430 

1 

530 

530 

1400 

1 

500 

500 

1300 

5 

400 

2000 

1210 

1 

310 

310 

1200 

7 

300 

2100 

1155 

1 

255 

255 

1100 

71 

200 

14200 

1050 

83   Total 

150 

12450 

1045 

3  teachers 

145 

435 

Approxi-   1000 

762   above  = 

100 

76200 

mate      950 

108   1050 

50 

5400  118810 

Median  =  900 

196    196 

0 
50 

0 

850 

112 

-5600 

825 

2 

75 

150 

800 

130 

100 

13000 

770 

1 

130 

130 

750 

133 

150 

19950 

715 

4 

185 

740 

700 

164 

200 

32800 

650 

145 

250 

36250 

600 

136   Total 

300 

40800 

550 

20   teachers 

350 

7000 

500 

110   below  = 

400 

44000 

400 

1    958 

500 

500  200920 

2)2204 

«*» 

1102 

The  deviations  have  been  calculated  with  reference  to  the  median 
in  this  particular  example  because  this  was  the  central  tendency 
used  in  the  survey.  But  it  is  apparent  that  if  the  modal  salary  of 
$1000  as  denoted  by  the  largest  group  of  762  cases  were  taken,  the 
discrepancy  between  the  deviation  measures  on  the  two  sides  would 
bo  much  greater. 


Measures  of  Deviation  or  Dispersion     161 


*l200-p 
1 160  4- 
1120- 

toao- 

10404- 

1000 

960, 

920 
860 
640 
800' 


Hohoktft 


760 


M 


680 

64  0 


560 
520 
480- - 
440 


Springfield 


Qayonne 


Youngstown 
Fort    Wayne 


Da   Moines 


Passaic  Somerv/t/e — Springfield — Lawrence — Hew  Bedford~--Eransvi/le-Du/uth  - 


Utica- 


■Waferbury- 


-Holyokt- 


•  Lynn.- 


Paw  tucket  ,      .  Trenton  Terre  Haute,,     ...   Kansas  Cy 

Elizabeth,  Wichita  J 

£.  St.  Louis 


—Schenectady Saginaw St.Joseph Wi/hes-  Barre 

Harris  bury 


Altoona 


Manchester 


Reading 


South  Bend 


Fig.   21.  — (Jraphod    Hoi, hill    Table   of    Moan    Annual    Salaries   Paid 
montarv  Teachers  in  Certain  Cities. 
(J.   K.   Bobbin  :    Elvmmturu  School  Journal,   15:  t".  ! 


162  School  Statistics  and  Publicity 

In  this  connection  it  should  be  noticed  that  a  Bobbitt 
table  may  be  really  a  skew  distribution.  This  skewness 
will  not  be  apparent  at  a  glance  because  the  table  will 
show  just  as  many  lines  or  cases  between  one  quartile 
and  the  median  as  between  the  other  quartile  and  the 
median.  It  needs  inspection  to  see  whether  cases  listed 
on  separate  lines  are  really  the  same  in  size  or  approxi- 
mately so.  Professor  Bobbitt  removes  this  difficulty  by 
graphing  his  results.  (See  page  101.)  The  crowding  in 
of  the  cases  below  the  median  shows  that  the  devia- 
tion is  on  the  average  much  less  on  that  side  than  on 
the  upper.  The  same  is  true  of  the  graph  given  in  Fig- 
ure 21. 

VIII.    WHICH  MEASURE  OF  DEVIATION  TO  USE  IN  A  GIVEN 
DISTRIBUTION 

The  choice  of  the  measure  of  deviation  depends  upon 
the  measure  of  central  tendency  selected.  If  the  latter  is 
the  average,  then  the  deviation  should  be  expressed  by 
the  average  deviation.  If  the  median  is  used,  deviation 
should  be  expressed  by  the  Q,  although  the  average  de- 
viation may  be  employed.  If  for  any  reason  extreme 
variations  are  to  be  emphasized,  the  extreme  range 
measure  or  the  standard  deviation  shoul^l  be  employed. 
In  a  skew  distribution,  if  the  median  is  used,  the  quartile 
deviation  or  the  average  deviation  for  each  side  should  be 
given  separately ;  if  the  mode  is  used,  the  average 
deviation  or  the  median  deviation  for  each  side  should  be 
given  separately. 


Measures  of  Deviation  or  Dispersion     1G3 

EXERCISES 

1.  Calculate  all  the  measures  of  deviation  or  dispersion  you  can 
for  each  of  several  of  the  distribution  tables  used  in  previous  exercises. 
For  this  use  the  measures  of  central  tendency  chosen  in  the  exercise 
on  page  148. 

2.  For  each  of  the  distributions  used  in  the  exercise  above,  which 
measures  of  deviation  are  preferable?     Why? 

REFERENCES   FOR   SUPPLEMENTARY   READING 

King,  W.  I.     Elements  of  Statistical  Method,  Chapter  XIII. 

Rugg,  H.  O.     Statistical   Methods   Applied  to   Education,    pp.    149- 

173  and  178-180. 
Thorndike,  E.  L.     Mental  and  Social  Measurements,  pp.  46-50  and 

Chapter  VI. 


CHAPTER   VII 

MEASURES    OF   RELATIONSHIPS 

So  far  we  have  for  the  most  part  considered  distribu- 
tions of  measures  as  wholes  and  a  single  distribution  at 
a  time.  But  one  of  the  greatest  values  of  statistical  method 
is  the  ease  with  which  it  makes  possible  the  study  of 
relationships,  including  those  between  separate  distribu- 
tions. Since  things  have  no  meaning  except  through 
their  connections  with  other  things,  statistical  method 
must  bring  out  such  relationships  very  clearly.  It  was, 
of  course,  impossible  to  cover  the  previous  topics  in 
this  book  without  dealing  with  relationships.  But  it 
is  now  advisable  to  give  these  connections  a  special  and 
separate  treatment. 

I.   RELATIONSHIPS   INSIDE   OF   ONE   GROUP 

Discrete  Series.  As  soon  as  a  Bobbitt  table  or  cen- 
tral tendency  and  the  deviation  for  any  distribution 
are  given,  by  any  of  the  various  ways-Jor  calculating 
these  measures,  a  relationship  is,  of  course,  indicated. 
But  in  general,  inside  of  one  distribution,  relationship  is 
most  easily  shown  by  some  form  of  the  bar  graph. 

If  every  item  is  kept  separate,  each  should  be  repre- 
sented by  a  separate  bar.  All  bars  should  be  the  same 
width  and  differ  only  in  length.  This  length  indicates 
the  measure  of  the  case. 

164 


Measures  of  Relationships  1G5 

The  graphing  of  the  Bobbitt  table  in  Figures  8  and  9  is  an  example 
of  this.  The  bars  in  Figure  8,  of  course,  may  run  vertically  if  pre- 
ferred. 

In  such  a  table  the  cases  are  arranged  from  high  to  low,  and  the 
tops  of  the  bars  or  the  curve  for  the  graphic  presentation  will  have  a 
general  downward  or  upward  direction,  according  to  the  end  from  which 
it  is  viewed.  But  there  are  some  distributions  in  which  the  cases  come 
in  a  time  sequence  or  in  an  alphabetical  order  so  that  the  tops  of  the 
bars  or  the  curve  jog  up  and  down.  This  would  be  the  case  with  the 
number  of  children  in  school  by  years  or  months  in  a  city  of  fluctuat- 
ing population,  if  the  base  scale  represented  the  calendar  months  or 
years  in  succession.  In  this  instance,  the  central  tendency  and  meas- 
ure of  deviation  would  have  to  be  calculated  and  given  separately, 
or  else  shown  with  the  same  data  rearranged  as  a  Bobbitt  table  or 
ordinary  grouped  distribution. 

Continuous  Series.  If  there  is  more  than  one  case  to 
an  item,  that  is,  if  the  items  are  grouped,  the  length  of 
each  bar  will  represent  the  number  of  cases.  This 
amounts  then  merely  to  drawing  the  surface  of  frequency 
with  bars  representing  groups  which  may  or  may  not  be 
adjacent  to  each  other.1  The  tops  of  the  ends  of  these 
bars  form  the  broken  line  or  "curve"  which,  united  with 
the  base  line,  makes  up  the  surface  of  frequency. 

If  the  distribution  table  is  to  be  written  or  printed 
without  graphing,  the  relative  sizes  of  the  groups  may 
frequently  be  brought  out  more  forcibly  by  turning  the 
numbers  into  percentages  of  the  whole  number  of  cases. 
Of  course,  the  ratio  between  47  cases  of  magnitude  6  in  a 
total  of  470  cases  is  exactly  the  same  as  the  ratio  of  10 
per  cent  to  100  per  cent.  But  the  ordinary  man  is  used 
to  thinking  in  terms  of  percentages  and  can  grasp  the 
relationship  much  more  quickly  that  way.  Graphing 
will  show  this  relationship  as  easily  from  the  original 
figures  as  from  percentages,  so  there  is  no  need  for  the 

iSeep.  Ill 


166 


School  Statistics  and  Publicity 


change  to  percentages  where  graphing  is  to  be  done,  in 
the  case  of  one  distribution  taken  by  itself. 

II.    SIMPLE    RELATIONSHIPS     BETWEEN    DIFFERENT    DIS- 
TRIBUTIONS 


If  two  different  distribu- 
tions have  been  made  up  on 
the  same  scale,  they  can  be 
conveniently  shown  in  bar 
graphs,  as  in  Figure  22. 

It  may  be  noted  there 
that  the  tops  of  the  two  dif- 
ferent sets  of  bars  give  two 
"  curves."  This  device  will 
not  be  of  service  in  compar- 
ing more  than  two  distribu- 
tions at  a  time,  but  several 

ria.  22.  —  Device  tor  (  omparmg 
Two    Different   Distributions    Made   more         '  Curves  "      may      be 

Gra°hs  th°  Same  SCal°  With  Bar  drawn  Provided  they  do  not 

The  whole  bars  represent  the  total  en-    Overlap  tOO  much.       It  IS  CUS- 

roiiment,  while  the  shaded  portions  reprc-  tomary   to   represent  curves 

sent   retarded   children   in   the   grades   at 

Memphis,  about  1908:    (From  Laggards  in  alone      by     different     COnVen- 

Our  Schools   by  L.  1'.  Ayres,  page  39,  by  ,  .  -. 

permission  of  Russell  SageFoundation)  tlOnal    Signs    as  '. 


No.  I- 
No.  II  - 
No.  Ill 


No.  IV.  .  .^ 

No.  V   -.-.-, 
No.  VI 


Figure  23  is  a  good  example. 

However,  it  is  often  very  difficult  to  grasp  comparisons 
between  two  different  distributions  using  the  same  scale, 
unless  the  number  of  cases  for  each  magnitude  is  put 
on  the  same  basis  as  regards  the  rest  of  its  distribution. 


Measures  of  Relationships 


167 


90 

80 
70 

60r 

50 
40 
30 

20 


Highest  ^ 
Average  LuTTE 

Lowest  J 
Average  for 
schools  in  22 
cities 


This  is  most  easily  done  by  changing  the  number  of  cases 
in  each  step  to  the  proper  percentage  of  the  whole.  That 
is,  one  would  not  give  the  achievements  in  composition 
for  several  school  grades 

,  .  .    .  RESULTS     OF   SPELLING   TESTS 

by    merely    citing    the  percentage  of  words  spelled  cor- 
number    of    children    in  Rectly    by  grades 
each  grade  making  the  Percen+ 
various  scores.     Instead, 
he    would    give    tables 
presenting  the  percent- 
ages of  children  making 
each  score  in  each  grade. 

Table  18  on  page  154 
employs  percentages  for 
making  comparisons  be- 
tween cities  on  salaries 
of  teachers,  by  groups. 

A  good  way  to  compare 
several  distributions  is  to 
place  the  surfaces  of  fre- 
quency made  up  from 
the  percentage  tables  one  FlG  2:] 
above  the  other.  This, 
of  course,  may  be  used      Thia  char*  *v™*n<*  th"  ™nw 

'  J  poorest    to   the   best  room   tested  m   each   tirade 

Only    When     they    are     all     in    spelling    Butte,    Montana.       For    example, 

the    poorest     seeond-Krade    room    averaged    7:i. 
Same     The     average     for    the     wliole     city     is    repre- 
sented   by    the    doted    line,  while    the    average 
LdSt,     f()r    twenty.two    cities    is     represented     l>\     the 
they     mUSt     be     Centered     heavy  line  at  70.      (From  Butt,;  Montana,  Sur- 
,,         ,,,  .        vey,  pane  72) 

exactly   (the   centers    m 

the  same  vertical  line),  or  else  the  differences  between  the 
central  tendencies  must  be  drawn  to  scale  accurately. 
Figure  24  is  a  good  example. 

Professor    Bobbitt   in   the   San    Antonio   Surrey   used 


v         1     1 

hi    \    \ 

// 

V 

m 

-- 

\ 

— 

— 

— ■ 

• — 

1 

2    3    4    5    6    7    8   Grades 

Use  of  Curves  for  Comparing 
More  Thau  Two  Distributions. 
This   chart    represents   the    range    fr 


applied     to 
scales.      In 


the 
this 


Scores        0 


2  3  4  5  6  7 

Median  Scores 

is:      Yin 


8  9 


Scores:    01   23456789 


Fig.   24. — Graph   for  Comparing    Itolatod   Distributions,   Using  the  Com 
position  Scores  in  Sail   Lake  City. 
(From  Salt  Luke  City  Surrey,  pace  141) 


168 


SPELLING     ABILITY    BY    GRADES 


100 


00 


Jjiad&JJL 


70 

60 

Q 

50 

M 
40 

0, 
30 

20 


M4,i22 

2?      £ 24. 


fir^oTV 


-GradfliZ 


/g 


?7>Z*I 


26     9   ,„  16 


Fig.   25.  —  Device   for   Graphing  Bobbitt   Table.-  from   Different    Distribu- 
.  t ions  on  Rela  led  Scales. 

Each  wide  column  represents  (he  achievement  in  spelling;  of  (he  ward  schools  for 
that  grade,  on  the  Ayres  scale.  Each  number  represents  a  ward  school,  and  the  height 
on  the:  scale  shows  the  achievement  of  that  grade  in  that  school.  The  ipiartile  and 
.median  lines  are  shown.      (Adapted  from  the  Sun  Antonio  Survey,  page  lu.>) 

169 


170  School  Statistics  and  Publicity 

another  device  for  making  comparisons  between  different 
distributions  that  had  been  made  up  as  Bobbitt  tables. 
He  drew  their  quartiles  and  medians  as  horizontal  lines 
across  vertical  columns,  allowing  each  distribution  one 
column  and  adjusting  the  horizontal  lines  on  the  vertical 
ones  as  scales.  Figure  25  gives  part  of  his  device  for 
showing  achievements  of  different  grades  in  spelling. 

This  is  a  very  convenient  graph,  for  it  shows  a  good 
many  relationships  at  a  glance,  such  as  the  variations  in 
central  tendencies,  dispersions,  etc.  By  observing  only  the 
lines  belonging  to  the  medians,  a  curve  may  be  read 
across  the  page.  That  is,  a  little  practice  will  enable 
one  to  see  three  curves  on  the  chart,  the  upper  rep- 
resenting quartile  3 ;  the  middle,  the  median ;  and  the 
lower,  quartile  1. 

III.    COEFFICIENT    OF   VARIABILITY    OR   DISPERSION 

A  serious  difficulty  arises  when  we  try  to  compare 
two  distributions  as  regards  the  amounts  of  dispersion 
unless  they  have  about  the  same  central  tendencies.  If 
the  central  tendencies  are  widely  different  and  the  abso- 
lute number  of  units  of  dispersion  is  the  same,  the  real 
or  relative  amounts  of  dispersion  are  widely  different. 

For  example,  suppose  a  superintendent  is  studying  the  way  his 
high  school  teachers  mark  pupils.  He  gets  500  or  more  marks  given 
by  each  teacher.  He  finds  that  two  teachers  have  the  same  median 
in  their  marks,  80  on  a  numerical  scale.  But  one  teacher  has  a  Q  of 
5  and  the  other  a  Q  of  10.  In  the  first  case,  the  typical  dispersion  is 
6|  per  cent  of  the  median;  in  the  other  it  is  12'  per  cent.  Again, 
a  median  of  90  with  a  Q  of  5  would  be  markedly  different  from  a 
median  of  80  with  a  Q  of  5.  In  the  former,  the  variability  would 
be  5^  per  cent  and  in  the  latter,  6}  per  cent,  although  they  varied 
10  points  in  the  central  tendency. 

Consequently,    in   comparing   distributions   or   groups 


Measures  of  Relationships  171 

on  variability,  experienced  persons  compare  them  through 
percentages  of  variability  or  dispersion.  The  percentage 
of  dispersion  is  usually  called  the  coefficient  of  variability. 
By  formula  it  is  simply : 

r^     ax  •     x.    e^T    ■  vn        Measure  of  Deviation 
Coefficient  of  Variably  .      MeasureofType      ,  pointed 

off  as  per  cent. 

Thus,  for  the  distribution  on  page  134,  the  calcula- 
tion is : 

Coefficient  of  Variability  =  §-  =   2.85  (from  p.  153)  =  23  g 

M     11.98  (from  p.  135)  /o' 

Professor  Haggerty  in  his  study  of  arithmetic  in 
twenty  Indiana  cities  J  has  an  interesting  graph  for  com- 
paring variation  in  two  distributions.  He  had  figures 
for  each  school  grade  on  the  Courtis  Tests  in  both  attempts 
and  rights,  thus  getting  two  distributions  of  twenty  cases 
each.  So  he  made  a  scale  for  the  attempts  and  one  for 
the  rights  on  each  grade,  with  the  medians  centered  and 
opposite.  Thus  a  line  joining  these  two  centers  was 
horizontal.  A  given  city  could  be  indicated  by  finding 
its  position  on  each  of  the  two  scales  and  joining  these 
points  by  a  line.  If  the  given  city  varied  as  did  the  whole 
group  of  cities,  its  line  was  parallel  to  the  original  line 
joining  the  medians.  If  it  did  not  vary  that  way,  its 
line  would  slant,  the  end  farthest  from  the  median  line 
indicating  that  it  varied  more  in  that  respect  than  in  the 
other. 

Thus  in  Figure  26,  for  the  fifth  grade  in  addition,  the  upper  heavy 
line  represents  the  work  of  the  city  of  Bloomington  and  shows  at  a 
glance  that  the  fifth  grade  in  Bloomington  attempted  9  problems  in 
addition  and  got  about  5 \  of  these  right;    that  in  both  attempts  and 

1  Haggerty,  M.  E. :  "Arithmetic:  A  Cooperative  Study  in  Edu- 
cational Measurements,"  Indiana  University  Studies,  No.  27 


14 
13 

7 

12 

. 

II 

► 

6 

10  ■ 

/ 

B/oorningfon. 

9 

/ 

5 

e  ' 

4 

7 

i 

Median 

6 

■ 

20  Ind  Cities. 

3 

5 

4 

i 

2 

3- 

2 

- 

1 

1 

A      R 

Fifth  grade  addition  Bloomington  Ind.  compared 
with  standard  for  20  Ind.  cities,  Haggerty. 

Vw.    20. —  Graphic    Device    for    Comparing    Variation    in    Two    Different 

1  Hstrilmtion.s. 

The  sonic  for  attempts  in  the  Courtis  Tests  from  twenty  cities  is  shown  on  the  left, 
centered  wit)]  the  scale  for  rights  on  the  right.  The  slanting  line  represents  the  achieve- 
ment of  Bloominirton,  the  slant  upward  indicating  that  it  is  relatively  higher  in  rights 
than  in  at  tempt.s. 

172 


Measures  of  Relationships  173 

rights  it  exceeded  the  standards  set  by  the  twenty  Indiana  cities  as 
a  group  (these  standards  are  6.5  and  3.6  respectively);  also  that  its 
achievement  in  rights  was  better  than  its  achievement  in  attempts. 

EXERCISE 

Which  of  the  two  distributions  given  in  Exercise  2,  page  122, 
is  the  more  variable  and  just  how  much?  Precisely  how  do  you  reach 
your  conclusion?  For  this,  use  the  central  tendencies  and  measures 
of  deviation  previously  calculated. 

IV.    CORRELATION 

Meaning  of  Correlation.  Sometimes  it  is  very  advan- 
tageous to  be  able  to  show  accurately  and  briefly  the 
general  relation  between  two  distributions  that  have 
some  common  element  or  have  been  tested  on  the  same 
thing.  Thus  it  may  be  desired  to  get  the  relation  between 
the  two  distributions  obtained  by  testing  the  same  group 
of  cases  on  two  different  tests.  For  example,  one  may 
wish  to  know  if  a  group  of  cities  rank  the  same  way  on 
excellence  of  schools  that  they  do  on  per  capita  school 
costs. 

Now  it  is  evident  in  all  such  cases  that  the  chances  are 
very  much  against  any  situation  where  the  cities  would 
rank  exactly  the  same  or  exactly  opposite  in  both  lists. 
Some  will  fall  in  exactly  the  same  places,  others  exactly 
opposite,  and  still  others  will  change  indiscriminately. 
Consequently  it  is  desirable  to  have  some  way  of  showing 
the  extent  to  which  the  individual  cases  generally  keep  the 
same  relative  positions  in  the  two  distributions  (thai  is, 
first  in  each,  second  in  each,  last  in  each,  etc.  I,  or,  putting 
it  another  way,  the  extent  to  which  the  two  distributions 
are  cprrelated  with  each  other. 

For  a  concrete  example,  let  us  take  the  following  data 
on  Cleveland  ward  schools,  accumulated  during  the  sur- 


174  School  Statistics  and  Publicity 

vey there.1  The  records  of  eighteen  schools  for  the  same 
grade  on  two  qualities,  which  for  our  purposes  we  may  call 
the  A  test  and  the  E  test,  are  given  in  Table  19. 

Table  19.     Correlation  Table  Using  Data  from  Eighteen 
Cleveland  Schools 


School 


Record  in 

Record  in 

A  test 

E  test 

32.8 

8.8 

28.3 

10.0 

28.0 

7.5 

26.3 

6.8 

26.1 

6.5 

25.5 

7.5 

25.0 

7.0 

24.6 

7.0 

24.0 

6.9 

23.0 

5.9 

22.9 

6.7 

22.6 

8.5 

21.9 

6.2 

21.9 

5.3 

21.7 

7.1 

21.3 

6.6 

20.8 

7.1 

19.4 

5.2 

Brownell  .  . 

Clark       .  .  . 

Marion    .  .  . 

Detroit    .  .  . 

Fullerton  .  . 

Sackett    .  .  . 
North  Doan 
Bolton 

East  Boulevard 

Gilbert     .  .  . 

Rosedale  .  . 

Landon    .  .  . 

Lawn       .  .  . 

Walton    .  .  . 

Gordon  .  . 

Sibley      .  .  . 

Waverly  .  . 

Halle        .  .  . 


For  our  purposes  it  is  not  necessary  to  know  which  grades 
were  used,  what  the  tests  mean,  or  how  they  were  figured. 
With  the  records  as  given,  the  main  question  is  :  How  did 
the  rankings  of  the  schools  on  the  two  tests  correspond  ? 
That  is,  did  each  school  (or  the  majority  of  the  schools) 

1  From  some  material  turned  over  to  the  author  to  be  used  as 
practice  work  in  his  class  on  Statistical  Methods  Applied  to  Educa- 
tion at  the  University  of  Chicago,  summer  quarter  of  1915. 


Measures  of  Relationships 


175 


Rating 


Rating 
35 


\ 

25 

\ 

20 

15 
10 

- 

- 

- 

5 

A  Test 


E   Test 


Fig. 


:  Z  I  S  3  S  I  i  .»  g  q  §  |  g  g  -       - 

)Oioi..«)ZoiuO(tjj>a<n>I 
—  Graph  to  Show  Correlation  Based  uii  Uaia 


,1  Tabic  l'J. 


176  School  Statistics  and  Publicity 

occupy  the  same,  or  approximately  the  same,  relative 
position  from  the  best  on  one  test  that  it  did  on  the 
other  ? 

Graphic  Methods  of  Showing  Correlations.  On  look- 
ing at  the  figures  we  note  that  the  poorest  school  on  the 
A  test  is  also  poorest  on  the  E  test.  But  the  best  on  the 
A  test  is  only  second  on  the  E  test.  The  second  best  on 
the  A  test  is  best  on  the  E  test.  Similar  observations 
might  be  continued  without  arriving  at  any  idea  of  the 
relation  existing  between  the  distributions  as  wholes. 

But  let  us  make  two  horizontal  scales,  one  on  each  side 
of  the  page,  running  from  0  to  35,  and  plot  the  two  tables 
with  the  names  of  the  schools  on  the  base  line,  as  in 
Figure  27. 

From  this,  it  is  apparent  that  in  general  the  schools 
have  about  the  same  relative  positions  on  the  two  tests, 
only  there  are  some  jogs  in  the  lower  curve  showing  that 
the  relationship  does  not  hold  for  every  case.  If  it  did 
hold  absolutely  for  every  case,  the  second  curve  would 
be  parallel  to  the  upper. 

Another  way  to  study  the  relationship  is  to  draw  a 
scale  for  each  test  on  opposite  sides  of  the  paper,  prefer- 
ably making  the  two  scales  about  the  same  height,  and 
drawing  a  straight  line  across  from  the  correct  place 
on  each  scale  to  represent  the  performance  of  each  school. 
This  is  merely  an  extension  of  the  device  used  by  Pro- 
fessor Haggerty.1  If  a  school  did  as  well  on  one  test  as 
on  another,  its  line,  of  course,  is  parallel  to  the  lines  of 
most  of  the  other  schools.  If  it  did  not,  the  line  will 
slant,  depending  upon  which  test  it  made  the  best  record 
on.  The  Cleveland  data  are  shown  by  this  device  in 
Figure  28.     Observe  that  many  of  the  lines  are  roughly 

!Seep.  172 


Measures  of  Relationships 


177 


parallel  but  that  a  fair  number  of  them  cut  across  each 

other  and  do  not  follow  the  general  trend  of  relationship. 

Showing  Correlation  by  Shifts  of  Cases  within  Parts 

of    the    Distribution.     One    very    rough    way    to    show 

correlation  is  to  divide  each   A  ^ 

,     .  ,.         ^       A-Tgst  E-Tert 

group  by  its  median,     then 

look  at  each  case  to  see  if  it  40 
is  in  the  same  half  (upper  or 
lower)  in  the  second  distri- 
bution as  in  the  first.  Call  35 
the  upper  half  "  plus  "  and 
the  lower  half  "  minus  "  in 
each  distribution.  Then  get 
the  percentage  of  like  signs. 
This  single  figure  which 
expresses  the  general  rela- 
tionship between  the  two 
distributions  is  called  the 
"  coefficient  of  correlation." 

Graphically  this  relation- 
ship may  be  represented 
easily  by  writing  the  cases 
in  one  distribution  in  two 
colors,  one  for  each  half,  say 
black  for  the  upper  and  red 
for  the  lower.  Then  keep 
these  same  colors  for  each 
case  in  the  second  distribu- 
tion, no  matter  where  the  case  falls.  Suppose  for  example 
that  all  cases  in  the  upper  half  in  the  first  distribution 
appear  black  and  all  in  the  lower  half  appear  red.  In 
the  second  distribution,  if  the  cases  fall  exactly  opposite, 
the  upper  half  will  be  red  and  the  lower  black.     If  there 


Fig.  2S.  - — Graphic  Representation 
of  Correlation  between  the  Results 
in  the  A  Test-  and  those  in  the 
E  'Pest    for    Eighteen     Cleveland 

Schools. 

Each  line  represents  a  school.      (From 
data  in  Table  19,  page  174.) 


178 


School  Statistics  and  Publicity 


is  a  good  deal  of  shifting,  there  will  be  some  red  and  some 
black  in  each  half. 

For  the  Cleveland  data,  this  device  is  shown  in  Table 
20,  the  upper  half  of  the  first  distribution  being  denoted 
by  all  capital  letters,  and  the  lower  half  by  small  letters, 
each  case  retaining  its  own  type  when  it  appears  in  the 
second  distribution. 


Table  20.    Like  Signs  Table   for  Correlation  of  Stand- 
ings of  18  Cleveland  Schools  on  Two  Tests,  Form  I 


Median 


Rank  and  name  of  school 

Rank  and  name  of  school 

for  A  test 

for  E  test 

1 

BROWNELL 

1 

CLARK 

2 

CLARK 

2 

BROWNELL 

3 

MARION 

3 

London 

4 

DETROIT 

4.5 

MARION 

5 

FULLERTON 

4.5 

SACKETT 

6 

SACKETT 

6.5 

Gordon 

7 

NORTH    DOAN 

6.5 

Waverly 

8 

BOLTON 

8.5 

NORTH    DOAN 

9 

EAST    BOULEVARD 

8.5 

BOLTON 

10 

Gilbert 

10 

EAST   BOULEVARD 

11 

Rosedale 

11 

DETROIT 

12 

London 

12 

Rosedale 

13 

Lawn 

13 

Sibley 

14 

Walton 

14 

FULLERTON 

15 

Gordon 

15 

Lawn 

16 

Sibley 

16 

Gilbert 

17 

Waverly 

17 

Walton 

18 

Halle 

18 

Halle 

Twelve  of  the  cases  are  in  the  corresponding  half  in  the 
two  distributions,  so  the  coefficient  of  correlation  is 
roughly  f|  or  66|  per  cent.1 

1  This  is  not  the  real  figure  for  this  coefficient  nor  is  the  procedure 
used  a  strictly  accurate  one.  But  both  are  sufficiently  accurate  for 
many  of  the  correlations  that  the  superintendent  will  need  until  he 


Measures  of  Relationships 


179 


In  using  this  method  there  must  be  as  many  ranks  in 
each  distribution  as  there  are  pairs  of  cases,  —  in  this 
instance  18.  If  two  schools  tie,  they  get  the  average 
rank  of  the  cases  occupied  by  them.  Note  that  Marion 
and  Sackett  on  the  E  test  are  given  4.5  each  because  they 
tied  for  the  4th  place  and  thus  occupy  the  4th  and  5th 
ranks,  or  average  4.5  each. 

Another  form  of  such  a  table  keeps  every  case  on  the 
same  line  for  the  two  distributions,  as  in  Table  21. 


Table  21.    Like  Signs  Table  for  Correlation  op  Standings 
of  18  Cleveland  Schools  on  Two  Tests,  Form  II 


Median 


Rank  and  name  of  school 

Rank  and  name  of  school 

for  A  test 

for  E  test 

1 

BROWNELL 

2 

BROWNELL 

2 

CLARK 

1 

CLARK 

3 

MARION 

4.5 

MARION 

4 

DETROIT 

11 

Detroit 

5 

FULLERTON 

14 

Fullerton 

6 

SACKETT 

4.5 

SACKETT 

7 

NORTH    DOAN 

8.5 

NORTH    DOAN 

8 

BOLTON 

8.5 

BOLTON 

9 

EAST   BOULEVARD 

10 
16 

East  Boulevard 

10 

Gilbert 

Gilbert 

11 

Rosedale 

12 

Rosedale 

12 

London 

3 

LONDON 

13 

Lawn 

15 

Lawn 

14 

Walton 

17 

Walton 

15 

Gordon 

6.5 

GORDON 

16 

Sibley 

13 

Sibley 

17 

Waverly 

6.5 

WAVERLY 

18 

Halle 

18 

Halle 

By  looking  at  the  right  hand  or  E  test  column,  it  is 
seen, at  once  that  there  are  three  cases  out  of  place  in  the 

has  studied  technical  statistics  much  more  thoroughly  than  is  here 
attempted. 


180  School  Statistics  and  Publicity 

upper  part  and  three  more  in  the  lower.  That  is,  12 
cases  out  of  18  or  66|  per  cent  of  the  cases  are  alike 
in  going  into  the  same  half  of  the  two  distributions. 

A  much  more  elaborate  device  of  the  same  nature,  but 
covering  two  grouped  distributions,  was  used  by  Pro- 
fessor Dearborn  to  show  the  relation  between  the  achieve- 
ments of  a  group  of  students  in  high  school  mathematics 
and  their  achievements  in  college  mathematics.1  He 
gave  each  student  a  number  and  placed  this  number  in 
the  proper  column  on  a  base  scale  carrying  the  grades 
from  60  to  100,  so  as  to  make  a  surface  of  frequency.  In 
the  high  school  mathematics  distribution,  all  numbers 
below  the  first  quartile  were  black ;  all  between  the  first 
quartile  and  the  median,  green ;  all  between  the  median 
and  the  third  quartile,  purple;  all  above  the  third 
quartile,  red.  Thus  there  were  four  equal  areas,  each 
represented  by  a  solid  color  (all  numbers  in  it  having  the 
same  color).  The  lowest  fourth  appeared  as  a  black 
mass,  the  second  fourth  as  a  green  one,  the  third 
fourth  as  a  purple  one,  and  the  upper  fourth  as  a 
red  one. 

With  the  same  number  and  color  to  represent  the  same 
student,  the  numbers  were  rmt  in  their  proper  places 
so  as  to  make  up  the  surface  of  frequency  for  achievement 
in  university  mathematics.  This  gave  #ie  second  dis- 
tribution. All  the  students  did  not  keep  exactly  the 
same  positions  that  they  had  in  the  first  distribution. 
Their  switching  around  showed  at  once  on  the  second 
distribution  because  the  colors  in  one  mass  were  not  all 
the  same  but  appeared  variegated.  The  switching 
around   in  colors  then  gave  a  rough   indication  of  the 

1  Dearborn,  W.  F. :  "School  and  University  Grades,"  in  Bulletin 
of  University  of  Wisconsin,  No.  368,  H.  S.  Series  No.  9,  p.  48 


Measures  of  Relationships  181 

relation  or  opposition  (correlation)  of  the  order  of  cases 
in  the  two  distributions. 

Coefficient  of  Correlation.  The  foregoing  ways  of 
showing  correlation  are  not  exact  enough  and  take  too 
much  space  to  show  the  desired  relation  with  the  best 
results  in  all  cases.  Consequently  there  have  been 
devised  various  ways  of  showing  this  relation  by  the 
single  figure  called  the  coefficient  of  correlation.  The 
calculation  of  this  coefficient  except  by  the  very  rough 
"  like  signs  method  "  is  generally  very  laborious,  and  the 
knowledge  of  what  the  results  mean  in  the  given  circum- 
stances is  very  hard  to  acquire. 

The  superintendent  should  not,  in  general,  enter  upon 
the  calculation  of  this  coefficient  without  much  more 
extensive  study  of  technical  statistics  than  can  be  given 
here.  Fortunately,  there  are  few  cases  where  the  su- 
perintendent really  needs  to  calculate  this  coefficient 
unless  he  is  working  on  a  thesis  in  some  university,  where 
he  will  be  shown  how  to  make  the  calculations.  However, 
he  does  in  his  reading  need  to  understand  what  the  co- 
efficient of  correlation  means.  Accordingly,  we  shall 
now  explain  clearly  what  it  is,  give  some  practical 
examples,  and  then  merely  append  two  ways  of  cal- 
culating it. 

Briefly,  the  coefficient  of  correlation  is  a  number  ex- 
pressing relationship  between  two  distributions,  based 
upon  the  changes  in  order  or  ranks  of  the  different  cases 
in  the  two  groups.  It  ranges  from  +  1  (or  +  100  per 
cent)  expressing  perfect  agreement  in  order,  through 
zero  expressing  only  accidental  or  no  relation,  to  -  1  (or 
-  100  per  cent)  expressing  perfect  opposition  in  order  or 
ranking.  In  perfect  relationship,  the  first  case  in  distri- 
bution one  will  be  the  first  case  in  distribution  two ;   the 


182  School  Statistics  and  Publicity 

second  case  in  distribution  one  will  be  the  second  case  in 
distribution  two,  etc.  In  perfect  opposition,  the  first  case 
in  distribution  one  is  the  last  case  in  distribution  two ;  the 
second  case  in  distribution  one  is  the  next  to  the  last 
case  in  distribution  two,  etc. 

There  is  marked  difference  of  opinion  among  statisticians 
and  investigators  as  to  the  significance  of  the  size  of  a 
coefficient  of  correlation.  Any  reader  of  educational  articles 
in  the  last  ten  years  which  employ  coefficients  of  correlation, 
will  have  noted  the  shifts  and  controversies  on  this  point. 
Coefficients  once  thought  large  enough  to  be  of  importance, 
are  now  considered  negligible.  The  size  depends  often 
upon  the  particular  method  of  calculation.  The  importance 
of  the  size  depends  upon  the  particular  method  used  for  the 
particular  distribution,  a  matter  about  which  the  authorities 
often  differ.  Probably  the  following  from  Professor  Rugg 
is  as  safe  as  anything  for  the  superintendent  in  his 
reading : 

This  definition  of  limits  (of  high  and  low  correlation)  depends 
largely  on  the  personal  experience  of  the  person  making  the  interpre- 
tation. For  example,  it  has  been  common  for  certain  educational 
investigators  to  interpret  arbitrarily  a  coefficient  of  .25  as  an  indi- 
cation of  "high"  positive  correlation,  and  one  of  .40  as  "very  high." 
Others  would  interpret  .25  as  very  low,  and  .50  as  "marked  "  or  "some- 
what high."  Certainly,  our  educational  conclusions  must  be  colored 
by  our  arbitrary  definition  of  such  a  coefficient.  The  experience  of 
the  present  writer  in  examining  many  correlation  tables  has  led  him 
to  regard  correlation  as  "negligible"  or  "indifferent"  when  r  is  less 
than  .15  to  .20;  as  being  "present  but  low"  when  r  ranges  from  .15 
or  .20  to  .35  or  .40;  as  being  "markedly  present"  or  "marked," 
when  r  ranges  from  .35  or  .40  to  .50  or  .60:  as  being  "high"  when  it 
is  above  .60  or  .70.  With  the  present  limitations  on  educational 
testing,  few  correlations  in  testing  will  run  above  .70,  and  it  is  safe 
to  regard  this  as  a  very  high  coefficient.1 

1  Rugg,  H.  O. :   Statistical  Methods  Applied  to  Education,  p.  256 


Measures  of  Relationships 


183 


Examples  of  Coefficients  of  Correlation.  The  following 
examples  of  coefficients  of  correlation  will  be  of  signifi- 
cance to  superintendents : 


Between  what 
items 

Coefficient  cf 
correlation 

Where  found 

Total  cost  per  pupil 

with 
Teaching  and  supervision 
(49  city  systems,  1902-3) 

+  .93 

Strayer,  G.  D. :  City 
School  Expenditures, 
p.  95 

Supervision 
with 
Repairs  (same) 

+  .15 

Same 

Payments  for  schools 

with 
Payments  for  interest 
on  city  debt 

-  .541 

Elliott,  E.  C:  Some 
Fiscal  Aspects  of  Public 
Education,  p.  85 

General  merit  of  teachers 

with 
Neatness  of  room 

+  .54 

Boyce,  A.  C. :  "  Methods 
for  Measuring  Teachers' 
Efficiency,"  in  1  Uh  Year- 
book, National  Society 
for  Study  of  Education, 
Part  II,  p.  68 

General  merit  of  teachers 

with 
General    development    of 
pupils 

+  .88 

Same 

General  merit  of  teachers 

with 
Discipline 

'    +  .79 

Same 

Ability  in  reasoning 

with 
Ability    in    fundamentals 
in  arithmetic 

+  .73 

Stone,  C.  W. :   Arithmeti- 
cal Abilities,  p.  37 

Ability  in  reasoning 

with 
Ability  in  addition 

+  .32 

Same 

184  School  Statistics  and  Publicity 

Calculation  of  Coefficient  of  Correlation.  The  original 
treatment  of  correlation  ended  with  the  preceding  exam- 
ples, but  several  readers  of  the  manuscript  recommended 
adding  one  or  more  simple  calculations  of  a  coefficient. 
For  reasons  which  will  not  be  clear  to  the  casual  reader, 
the  coefficient  of  correlation  calculated  from  the  Cleveland 
data  would  not  be  reliable  enough  to  be  worth  the  effort  to 
get  it.1  These  reasons  center  about  the  fact  that  the  orig- 
inal measures  were  averaged  from  a  number  of  individuals 
and  not  a  simple  measure  for  each  school.  But  as  it  would 
take  too  much  space  and  time  to  introduce  and  explain 
new  sets  of  data,  we  shall  assume  that  the  data  are  suit- 
able for  the  purpose  and  proceed  to  give  the  calculations. 
The  method  of  calculation  is  the  only  thing  we  care  for  here 
and  it  can  be  shown  as  well  with  the  Cleveland  data  as  with 
any  other.  The  reader  who  has  had  no  other  train- 
ing in  statistics  is  particularly  cautioned  against  attempt- 
ing to  calculate  coefficients.  The  examples  are  given 
solely  to  clear  up  a  reading  knowledge  of  correlation. 

1.    Spearman  Rank-Order  Method  2 

62  D- 

p    =  1  — ■ 

n{n2  —  1) 

2  =  sum  of 

D2  =  squares  of  differences  in  ranks  (must  be  same  number  of  ranks 

in  both  distributions) 

n   —  number  of  pairs  in  distributions 

p    =  coefficient  of  correlation 

'  Sec  J.  F.  Kelley:  Teachers'  Marks,  p.  88 

2  The  formula  here  given  is  only  one  factor  in  the  correct  formula, 
but  it  is  sufficiently  accurate  for  administrative  problems.  See  Thorn- 
dike  :  Mental  and  Social  Measurements,  p.  167 


Measures  of  Relationships 


185 


From  page  178  we  get  D-  as  follows: 


D 

D2 

1 

1.00 

Clark      

1 

1.00 

1.5 

2.25 

7 

49.00 

9 

81.00 

1.5 

2.25 

1.5 

2.25 

.5 

.25 

East  Boulevard 

1 

1.00 

Gilbert 

6 

36.00 

1 

1.00 

9 

81.00 

2 

4.00 

Walton 

3 

9.00 

8.5 

72.25 

Sibley 

3 

9.00 

10.5 

110.25 

Halle 

0 

0.00 

462.50 

p  =  1 


6X462.50 


=  +  .52 


18(324  -  1) 
2.    The  Pearson  Method 

The  Pearson  coefficient  of  correlation  is  calculated  by  the  following 
formula : 

2  (x     ij) 
Coefficient  of  Correlation  or  r= — -/; 

S  =  algebraic  sum 
x  =  deviations  in  one  distribution 
y  =  deviations  in  other  distribution 
x  •  y  =  product  of  deviations  for  corresponding  case. 


186 


School  Statistics  and  Publicity 


The  calculation  on  the  Cleveland  data  is  as  follows,  the  deviations 
being  taken  from  the  median : 


School 

Score   ■ 

X 

x2 

Score 

V 

?y2 

+xy 

—  xy 

Brownell  . 

32.8 

+9.3 

86.49 

8.8 

+  1.85 

3.42 

+  17.21 

Clark    .      . 

28.3 

+4.8 

23.04 

10.0 

+3.05 

9.30 

+  14.64 

Marion 

28.0 

+4.5 

20.25 

7.5 

+   .55 

.30 

+  2.48 

Detroit 

26.3 

+2.8 

7.84 

6.8 

-    .15 

.02 

-    .42 

Fullerton  . 

26.1 

+2.6 

6.76 

6.5 

-    .45 

.20 

-1.17 

Sackett     . 

25.5 

+2.0 

4.00 

7.5 

+   .55 

.30 

+    1.10 

N.  Doan  . 

25.0 

+  1.5 

2.25 

7.0 

+   .05 

.003 

+      .80 

Bolton . 

24.6 

+1.1 

1.21 

7.0 

+   .05 

.003 

+      .06 

E.  Blvd.    . 

24.0 

+   .5 

.25 

6.9 

-   .05 

.003 

.03 

Gilbert      . 

23.0 

-    .5 

.25 

5.9 

-1.05 

1.10 

+     .53 

Rosedale  . 

22.9 

-   .6 

.36 

6.7 

-    .25 

.06 

+     .15 

London 

22.6 

-    .9 

.81 

8.5 

+  1.55 

2.40 

1.40 

Lawn    . 

21.9 

-1.6 

2.56 

6.2 

-    .75 

.56    +   1.20 

Walton      . 

21.9 

-1.6 

2.56 

5.3 

-1.65 

2.72    +   2.64 

Gordon 

21.7 

-1.8 

3.24 

7.1 

+   .15 

.02 

-    .27 

Sibley  .     . 

21.3 

-2.2 

4.84 

6.6 

-    .35 

.12 

+     .77 

Waverly   . 

20.8 

-2.7 

7.29 

7.1 

+    .15 

.02 

-    .41 

Halle     .      . 

19.4 
Med.  23.50 

-4.1 

16.81 

5.2 

-1.75 

3.06 

+   7.18 

190.81 

Med.  6.95 

23.61 

+48.04 

-3.70 

2  xy 


44.34 


Vs  x-  ■  Vs 


=  +  .66     Ans. 


V190.81  V23.61 

(13.81)    (4.86) 

The  Pearson  coefficient  in  actual  work  is  often  figured  from  the 
guessed  average,  the  result  being  corrected  by  a  formula  given  in  the 
Thorndike  and  Rugg  references.  This  saves  tiny;  because  the  devia- 
tions will  thus  always  be  integers  or  steps  denoted  by  integers,  and  the 
handling  of  these  will  be  much  easier  than  when  deviations  with  deci- 
mals are  used  as  in  the  example  just  given. 


REFERENCES  FOR  SUPPLEMENTARY  READING 

King,  W.   I.     Elements  of  Statistical  Method,  Chapters  XIV,  XVI, 

XVII,  XVIII. 
Rugg,   H.   O.     Statistical  Methods  Applied  to  Education,   Chapters 

VII,  VIII,  IX. 
Thorndike,  E.  L.     Mental  and  Social  Measurements,  Chapters  X,  XI. 


CHAPTER  VIII 
SUPPLEMENT    ON    STATISTICAL   TREATMENT 

The  preceding  chapters  cover  most  of  the  problems 
which  the  superintendent  will  encounter  in  working  up 
his  school  statistics,  until  he  reaches  the  actual  presenta- 
tion to  the  public.  But  there  are  three  other  practical 
problems,  not  directly  connected  with  any  of  the  fore- 
going problems  nor  with  one  another,  which  will  be  of 
interest  to  him.     These  are : 

1.  How  may  one  insure  reliability  in  the  statistical  results  or  at 
least  know  about  how  accurate  they  are? 

2.  What  special  economies  may  be  employed  to  reduce  the  in- 
evitably large  labor  involved  in  statistical  calculations? 

3.  How  may  data  given  in  ranks  only  (for  example,  the  decisions 
of  judges  in  a  contest)  be  easily  combined? 

These  problems  will  now  be  discussed  in  order. 

I.    RELIABILITY    OF    STATISTICAL   RESULTS 

The  aim  throughout  this  book  has,  of  course,  been  to 
produce  reliable  results,  and  cautions  for  this  purpose 
have  been  included  in  many  places.  But  in  spite  of 
these,  the  reader  will  probably  wish  to  know  how  to 
avoid  the  numerous  errors  in  adding,  multiplying,  omit- 
ting figures,  etc.,  that  all  of  us  understand  are  liable  to 
creep  into  any  numerical  work. 

Variable  Errors.  The  simplest  way  to  regard  these  is 
to  distinguish  between  variable  and  constant  errors.     A 

187 


188  School  Statistics  and  Publicity 

variable  error  is  one  that  is  as  likely  to  occur  one  way  as 
the  other  and  so  the  various  results  will  offset  each  other. 
For  example,  in  adding,  the  average  person  is  as  apt  to  get 
results  too  large  about  as  often  as  he  gets  them  too  small 
and  in  about  the  same  amounts.  In  any  problem  in- 
volving many  additions,  these  errors  will  balance  each 
other  and  so  the  result  may  be  assumed  to  be  fairly  correct. 
The  principle  is  exactly  that  involved  in  the  common 
practice  of  discarding  the  last  decimal  place  by  adding  1 
to  the  preceding  place  if  this  decimal  is  5  or  more,  and 
throwing  it  away  altogether  if  it  is  less  than  5.  Or  it  is 
the  same  principle  as  that  behind  the  old  business  prac- 
tice of  settling  a  bill  to  the  nearest  five  cents,  —  paying 
$1.25  if  the  bill  comes  to  $1.23  or  only  $1.20  if  it  comes  to 
$1.22.  In  both  of  these,  it  is  assumed  that  in  the  long 
run  things  will  even  up  and  no  essential  injustice  or  in- 
accuracy will  result. 

Most  of  the  errors  in  ordinary  statistical  calculations 
are  variable  ones  and,  if  reasonable  care  is  given  the  work, 
they  may  then  be  ignored.  But  they  may  be  still  further 
removed  by  the  devices  of  utilizing  students  or  assistants 
to  check  each  other's  work,  or  of  going  through  each  pro- 
cess at  least  twice.  If  the  result  comes  out  the  same  each 
time,  it  may  be  safely  assumed  to  be  correct.  If,  after 
repeated  trials,  the  results  all  vary  slightly,  the  average 
of  them  is  safe  enough  for  practical  purposes. 

Constant  Errors.  A  constant  error,  on  the  other  hand, 
is  one  that  always  tends  in  the  same  direction.  The 
marks  in  deportment  given  healthy  boys  by  a  well- 
poised,  healthy  teacher  would  in  the  long  run  be  sub- 
stantially correct.  If  he  were  ill  one  month  and  cut 
them  very  low,  the  result  would  probably  be  offset  by 
correspondingly  high  marks  in  some  month  when  he  was 


Supplement  on  Statistical  Treatment     189 

feeling  well.  That  is,  his  variable  errors  would  offset 
each  other.  But  if  these  same  boys  were  under  a  nervous 
or  irritable  teacher,  his  errors  would  probably  all  tend 
in  the  direction  of  giving  them  very  low  marks.  That  is, 
he  would  have  a  constant  error.  Similar  constant  errors 
may  be  expected  in  such  cases  as  these :  reports  on 
number  of  tardies,  because  teachers  easily  forget  to  mark 
tardies  but  are  very  unlikely  to  mark  present  pupils  as 
absent  at  the  time  of  noting  tardies ;  reports  on  training 
of  teachers,  because  teachers  are  apt  to  try  to  make  their 
training  seem  as  extensive  as  possible  and  very  unlikely 
to  understate  it ;  itemized  and  verified  expense  accounts, 
which  usually  tend  to  be  less  than  the  actual  expenditure 
because  of  the  general  tendency  to  forget  to  put  down 
items,  and  the  absence  of  intent  to  be  dishonest. 

There  is  no  general  rule  for  avoiding  constant  errors, 
except  possibly  to  watch  for  fallacies  in  sampling.  Con- 
stant errors  are  never  negligible  but  "  skill  in  avoiding 
them  is  due  to  capacity  and  watchfulness  far  more  than 
to  knowledge  of  any  formal  rules."  l 

Weighting  Results.  Sometimes  the  attempt  is  made 
to  avoid  errors  by  weighting  certain  factors  to  give  them 
more  influence  in  determining  the  results.  This  should 
never  be  attempted  in  the  belief  that  it  will  give  more 
accurate  results,  except  by  one  especially  trained  in 
statistics.  "  Bowley  gives  a  rule  that  is  satisfactory 
for  most  cases  that  occur  in  practice ;  namely,  to  give 
your  attention  to  eliminating  constant  errors  and  not  to 
manipulating  weights."  2 

Estimating  Reliability  of  Results.  All  complete  treat- 
ments'of  statistical  methods  include  formulas  for  estimat- 

1  Thorndike,  E.  L. :   Mental  and  Social  Measurements,  p.  209 

2  Ibid.,  p.  212 


190  School  Statistics  and  Publicity 

ing  the  reliability  of  the  various  averages,  measures  of 
dispersion,  or  coefficient  of  correlation.  But  in  practi- 
cal school  statistics  there  is  no  need  for  these.  The 
superintendent  had  far  better  employ  his  time  in  seeing 
that  he  avoids  the  errors  in  collecting  and  manipulating 
his  data  that  common  sense  and  reasonable  care  would 
indicate.  Beyond  this,  he  needs  to  know  only  that,  if 
constant  errors  are  avoided,  accuracy  is  increased  by 
taking  more  measures  of  an  item,  or  samples  from  a 
distribution.  From  a  random  sampling  of  any  material 
number  of  cases,  accuracy  increases  as  the  square  root 
of  the  number  of  samples.  That  is,  400  samples  would 
double  the  accuracy  obtained  by  taking  only  100  samples. 

II.    SPECIAL   ECONOMIES   IN    CALCULATION 

Many  economies  have  been  indicated  in  connection 
with  the  particular  processes  to  which  they  pertain. 
But  there  are  several  others  that  merit  special  notice. 

Use  of  Cross-Section  Paper.  Plain  cross-section  paper 
is  very  valuable  for  saving  time  in  getting  figures  lined 
up  in  columns  and  in  keeping  horizontal  lines  straight. 
It  also  lessens  eyestrain.  The  kind  with  a  little  square 
just  large  enough  for  one  digit  is  best.  This  paper  is 
especially  valuable  for  saving  copying,  because  a  part 
of  a  calculation  that  is  correct  or  that  needs  to  be  shifted 
can  easily  be  cut  out,  removed  to  where  it  is  needed,  and 
quickly  pasted  into  line  there. 

Checking  in  from  Both  Ends.  The  old  device  of 
checking  addition  by  adding  the  column  up  and  then  down 
until  the  results  agree  is  very  helpful  for  insuring  accuracy. 
The  device  of  calculating  the  median  by  going  up,  and 
then  by  coming  down,  until  the  two  results  agree,  is' 
equally  serviceable. 


Supplement  on  Statistical  Treatment      191 

Use  of  Steps  in  Distributions  Involving  Large  Numbers. 
If  the  distribution  involves  large  numbers,  time  can 
usually  be  saved  by  employing  steps  instead  of  the  actual 
numbers  for  getting  the  average  from  the  guessed  average, 
for  getting  the  deviation  measures,  etc.  Thus,  if  one  is 
studying  superintendents'  salaries  grouped  by  hundreds, 
and  the  guessed  average  is  1500,  the  deviations  do  not 
have  to  be  given  in  hundreds.  The  1600  group  could  be 
counted  as  one  step  above ;  the  1400  group,  one  step 
below,  etc.  The  deviation  measure  finally  calculated  would 
then  be  in  steps ;  if  multiplied  by  $100,  the  value  of  a  step, 
the  real  deviation  measure  would  be  at  once  secured. 
This  method  obviously  saves  much  labor  in  copying  zeros. 
It  would  save  much  more  labor  in  multiplication,  if  the 
value  of  a  step  was  some  such  number  as  25,  5.5,  etc. 

One  Step  All  the  Way  Through.  It  is  much  easier  men- 
tally and  insures  accuracy  as  well,  to  perform  the  same 
process  all  the  way  through  at  one  time.  Thus,  all  data 
may  be  copied  to  the  end ;  all  multiplications  may  be 
made  at  one  step ;  all  additions  made  at  another ;  all 
squares  found  at  another,  etc. 

Calculating  Devices.  Various  tables  may  be  utilized 
to  save  time  in  multiplying,  dividing,  squaring,  or  ex- 
tracting roots.  These  are  mentioned  in  the  bibliography. 
The  writer  has  used  Crelle  for  years,  but  this  is  not  now 
obtainable  in  this  country.  The  short  tables  given  in 
Thorndike's  Mental  and  Social  Measurements  can  often 
be  utilized  by  employing  the  step  method  as  given 
above.  These  tables  are  not  at  all  difficult  for  even  school 
children  to  use.  Some  years  ago,  the  twelve-year-old 
son  of  a  friend  of  the  writer  regularly  used  the  Thorndike 
tables  in  computing  the  batting  averages  of  his  favorite 
baseball  stars. 


192  School  Statistics  and  Publicity 

Most  superintendents  will  be  able  to  get  the  use  of  an 
adding  machine  at  the  city  hall,  the  county  courthouse, 
or  some  bank,  if  the  school  system  does  not  have  one. 

Utilizing  Students  in  Calculations.  What  was  said 
about  the  value  to  students  of  collecting  data  (see  page 
86),  applies  even  more  to  practice  on  the  calculations 
involved  in  working  them  up.  Such  data  afford  the 
finest  sort  of  laboratory  material  for  the  upper  arithmetic 
and  high  school  mathematics  classes.  And  much  of  the 
work  is  very  valuable  for  clerical  practice. 

III.    COMBINING   DATA   GIVEN   IN   RANK    ORDER    ONLY 

The  school  administrator  often  faces  a  problem  which 
involves  a  summary  for  the  relative  merits  of  a  number 
of  items  that  have  been  graded  on  different  phases  with 
widely  varying  standards,  or  by  different  judges  with 
varying  standards.  Thus,  he  may  wish  to  find  the  vale- 
dictorian in  the  high  school  from  the  grades  given  by 
many  teachers.  He  may  wish  to  find  the  best  contestant 
in  a  debating  contest  from  decisions  given  by  several 
judges.  Or  he  may  wish  to  find  which  ward  school  is 
on  the  whole  the  best,  as  judged  by  results  from  standard 
tests  in  arithmetic,  composition,  spelling,  and  handwrit- 
ing, all  given  on  very  different  scales.  In  all  of  these 
he  must  find  the  relative  standing  or  rank  of  several 
people  or  schools.  How  can  he  combine  the  rankings 
given  from  one  judge  or  one  test  with  those  from  another 
judge  or  another  test? 

These  problems  may  be  conveniently  classified  for 
practical  purposes  as  being  of  five  types : 

1.    How  may  rankings  from  different  distributions,  having  the  same- 
number  of  ranks  and  being  considered  as  of  equal  value,  be  combined  ? 
For  example,  the  decisions  of  judges  in  a  contest. 


Supplement  on  Statistical  Treatment      193 

2.  How  are  tie  rankings  within  one  or  more  distributions  to  be 
treated  when  being  combined  with  rankings  from  the  other  distribu- 
tion? 

For  example,  one  judge  in  the  contest  gives  a  tie  vote  on  two 
contestants. 

3.  How  are  rankings  from  one  distribution  to  be  combined  with 
rankings  from  other  distributions  which  have  a  different  number  of 
ranks? 

For  example,  a  rank  from  a  class  of  12  in  chemistry  is  to  be 
combined  with  a  rank  from  a  class  of  20  in  English ;  or  rankings 
on  daily  grades  from  a  class  of  20  with  rankings  for  only  18 
who  took  the  examination. 

4.  How  are  rankings  from  different  distributions  to  be  combined 
when  it  is  desired  to  give  special  weight  to  one  or  more  of  the  dis- 
tributions? 

For  example,  it  is  desired  to  count  rankings  on  daily  grades 
three  times  as  much  as  rankings  on  examination,  to  secure  the 
rankings  for  final  grades. 

5.  How  is  absolute  unfairness  on  the  combinations  to  be  avoided? 

For  example,  in  the  author's  class  in  statistical  methods  at  one 
time,  tests  were  being  given  on  which  the  class  were  ranked. 
The  rankings  were  obtained  from  two  separate  distributions, 
one  of  rankings  on  speed,  the  other  of  rankings  on  accuracy. 
Speed  and  accuracy  were  counted  equal.  On  one  test,  a  stu- 
dent at  the  beginning  handed  in  a  blank  paper,  claiming  first 
rank  on  speed  and  being  willing  to  take  lowest  rank  on  accuracy. 
With  thirty  in  the  class,  he  counted  on  a  final  or  average  ranking 

of  ' — 1—  ,  or  15.5,  or  about  the  middle  of  the  class. 


These  will  be  discussed  in  order. 


1.    Distributions  Having  Same  Number  of  Ranks 

Tbis  is  managed  by  adding  the  various  ranks ;  then 
ranking  the  items  in  order  of  the  sums,  giving  the  one 
with  the  smallest  sum  a  final  rank  of  1,  the  one  with  the 
next  smallest  sum  a  rank  of  2,  etc. 


194 


School  Statistics  and  Publicity 


For  example,  take  the  illustration  of  the  judges  in  a  contest,  given 
on  page  12,  changing  some  of  the  figures  to  avoid  any  ties.  This 
would  keep  the  number  of  ranks  the  same  for  each  judge.  The  orig- 
inal marks  would  be : 


Contestant 

Marked  by 

Marked  by 

Marked  by 

Judge  A 

Judge  B 

Judge  C 

1 

55 

82 

89 

2 

95 

80 

94 

3 

70 

83 

92 

4 

85 

88 

93 

5 

71 

81 

91 

6 

84 

86 

95 

7 

80 

87 

96 

8 

75 

84 

97 

9 

77 

78 

98 

10 

60 

79 

90 

Changing  these  to  ranks  we  would  have : 


Contestant 

Ranked  by 

Ranked  by 

Ranked  by 

Sum  of 

Final 

Judge  A 

Judge  B 

Judge  C 

ranks 

rank 

1 

10 

6 

10 

26 

9 

2 

1 

8 

5 

14 

5 

3 

8 

5 

7 

20 

7 

4 

2 

1 

6 

9 

lor2J 

5 

7 

7 

8 

22 

8 

6 

3 

3 

4 

10 

3 

7 

4 

'     2 

3 

9 

lor2x 

8 

6 

4 

2 

12 

4 

9 

5 

10 

1 

16 

6 

10 

9 

9 

9 

27 

10 

If  the  judges  had  rated  the  contestants  by  rankings  in  the  first 
place,  the  result  could,  of  course,  have  been  obtained  much  more 
quickly. 

1  These  tie,  and  average  1.5  each.     See  page  195. 


Supplement  on  Statistical  Treatment      195 


2.    Tie  Rankings 

This  is  handled  by  changing  the  tie  rankings  so  as  to 
make  the  same  number  of  ranks  in  each  distribution, 
then  proceeding  as  in  1. 

For  example,  the  rankings  from  the  marks  of  the  judges  as  given 
on  page  12  would  be : 


Contestant 

Ranked  by 

Ranked  by 

Ranked  by 

Judge  A 

Judge  B 

Judge  C 

1 

8 

4 

5 

2 

1 

5 

2 

3 

6 

4 

3 

4 

2 

1 

2 

5 

6 

4 

4 

6 

3 

2 

1 

7 

4 

2 

1 

8. 

5 

3 

1 

9 

5 

5 

1 

10 

7 

4 

3 

Obviously,  these  could  not  be  combined  as  they  now  are,  for  two 
judges  used  only  five  ranks  and  the  other  one  had  eight,  so  that  rank 
1  does  not  mean  the  same  in  the  three  distributions. 

The  trouble  will  be  removed  by  changing  the  rankings  so  as  to 
make  each  judge  cover  the  entire  ten  ranks.  Thus,  in  the  original 
table  on  page  12,  Judge  A  has  given  Contestants  8  and  9  each  the 
rank  of  5.  This  means  he  has  really  made  them  take  up  ranks  5  and 
6,  or  average  5.5  in  rank.  This  leaves  Contestants  3  and  5  as  ranked 
by  him  to  occupy  ranks  7  and  8  or  average  rank  7.5  each.  Proceeding 
thus,  the  real  rankings  from  the  figures  on  page  12  would  be  those 
given  on  the  following  page. 

It  would,  of  course,  have  been  easier  to  require  the  judges  to  give 
no  tie  rankings,  but  if  they  should  insist  on  turning  in  tie  ranks,  the 
only  safe  procedure  is  that  given. 


196 


School  Statistics  and  Publicity 


Real  rank 

i 
Real  rank 

Real  rank 

Sum 

Final 
rank 

Contestant 

by 

by 

by 

of 

Judge  A 

Judge  B 

Judge  C 

ranks 

1 

10 

6.5 

10 

26.5 

10 

2 

1 

9.5 

5.5 

16 

5 

3 

7.5 

6.5 

7.5 

21.5 

7 

4 

2 

1 

5.5 

8.5 

2 

5 

7.5 

6.5 

9 

23 

8.5 

6 

3 

2.5 

2.5 

8 

1 

7 

4 

2.5 

2.5 

9 

3 

8 

5.5 

4 

2.5 

12 

4 

9 

5.5 

9.5 

2.5 

17.5 

6 

10 

9 

6.5 

7.5 

23 

8.5 

3.    Different  Number  of  Ranks  in  the  Distributions 

This  is  practically  the  same  as  the  problem  of  a  tie 
vote,  except  that  here,  instead  of  taking  the  average  of  a 
set  of  items  and  getting  the  average  rank  for  each,  one  may 
have  to  take  the  average  range  of  an  item  to  get  its  rank. 

For  example,  suppose  that  one  wishes  to  combine 
ranks  of  a  class  of  12  in  chemistry  with  those  from  a  class 
of  20  in  English.  The  ranks  in  chemistry  should  cover 
20.  This  will  make  one  rank  in  chemistry  cover  1.6 
ranks  in  English.  If  the  second  and  third  pupils  in 
chemistry  tied,  they  would  cover  ranks  3.2  and  4.8,  or 
average  rank  4  each. 

In  the  case  where  20  pupils  were  ranked  on  daily  grades, 
but  only  18  took  the  examination,  a  similar  procedure 
could  be  followed.  But  a  much  shorter  approximate 
result  might  be  obtained  by  making  the  18  pupils  get 
ranks  1  to  18  on  examination,  giving  19.5  each  on  examina- 
tion to  the  two  absent  pupils.  The  question  of  passing 
pupils  absent  from  examinations  would  have  to  be 
decided  entirely  apart  from  the  matter  of  ranks. 


Supplement  on  Statistical  Treatment      197 


4.    Weighting   Factors    Given   in   Ranks   from   Different 
Distributions 

The  way  to  do  this  is  to  arrange  the  factors  by  ranks, 
with  the  same  number  of  ranks  in  each  distribution.  Then 
multiply  the  factor  to  be  weighted  by  the  number  of 
times  it  is  desired  to  weight  it.  The  ranks  can  then  be 
added  as  in  previous  instances. 

For  instance,  suppose  pupils  have  been  rated  as  follows  on  exam- 
ination and  daily  grades : 


Pupil 

Ranks  on 

Rank  on 

daily  grade 

examination 

A 

1 

5 

B 

2 

2 

C 

3 

4 

D 

4 

3 

E 

5 

1 

If  it  is  desired  to  count  the  daily  grades  three  parts  and  the  examina- 
tion grade  one  part  in  determining  the  final  rating,  the  original  ratings 
for  daily  grades  should  be  multiplied  by  3  ;    thus : 


Pupil 

Weighted 

daily  grade 

rank 

Rank   in 

examination 

grade 

Sum  of 
ranks 

Final 
rank 

A 

3 

5 

8 

1.5 

B 

6 

2 

8 

1.5 

C 

9 

4 

13 

3 

D 

12 

3 

15 

4 

E 

15 

1 

16 

5 

i.    Avoiding  Absurdities  in  Combining  Ranks 

No  mechanical  device  or  general  advice  will  take  the 
place  of  common  sense  in  avoiding  absurdities  such  as 


198  School  Statistics  and  Publicity 

that  of  the  student  mentioned  on  page  193,  who  handed 
in  a  blank  paper.  In  this  particular  instance,  the  com- 
mittee of  students  appointed  to  consider  the  case  rightly 
decided  that  common  sense  and  fairness  demanded  that 
the  student  was  not  entitled  to  any  consideration  what- 
ever until  all  those  who  had  really  tried  to  do  something 
had  been  assigned  ranks.  Similarly,  the  matter  of  giving 
any  ranks  whatever  to  the  two  students  who  failed  to 
take  the  examination  in  the  instance  given  on  page  196 
would  have  to  be  decided  entirely  apart  from  the  mechan- 
ical problem  of  adjusting  the  ranks.  The  whole  matter 
of  ranking  might  be  left  until  these  two  had  taken  the 
examination.  But  as  such  a  delay  on  reporting  grades 
or  ranks  is  hardly  practicable  in  most  places,  it  would  be 
simpler  to  rank  all  the  others  and  later,  after  the  two  had 
taken  the  examination,  to  give  them  tie  ranks  that  would 
place  them  approximately  where  they  belonged.  Much 
energy  is  sometimes  wasted  in  trying  to  combine  rankings 
where  the  same  subjects  did  go  wholly  through  two  differ- 
ent sets.  Thus  a  student  was  worried  over  what  to  do 
with  an  experiment  with  fourteen  white  rats  where  one  died 
Li  training  so  that  he  had  to  train  a  new  fourteenth  rat. 
A.  junior  high  school  principal  studying  twenty  pupils  had 
only  nineteen  of  these  in  both  distributions  and  wished  to 
take  a  new  twentieth  one  for  the  second  test.  The  simplest 
and  safest  procedure  is  to  take  only  the  cases  that  appear 
all  the  way  through. 

EXERCISE 

Find  the  relative  standing  of  each  of  the  twenty  cities  for  rights 
in  arithmetic,  on  the  four  processes  combined,  from  the  data  on  the 
following  page. 


Supplement  on  Statistical  Treatment      199 

Rankings  of  Twenty  Indiana  Cities  on  Rights  in  Arithmetic 
for  the  Courtis  Tests  in  Fifth  Grade  Only  x 


Ranks 

City 

Addition 

Subtraction 

Multiplication 

Division 

1 

8 

6.5 

13 

14.5 

2 

19.5 

17.5 

20 

20 

3 

3 

4 

3 

6 

4 

12.5 

16 

15 

7.5 

5 

19.5 

19 

19 

19 

6 

5.5 

5 

9 

14.5 

7 

16.5 

12.5 

10.5 

9 

8 

16.5 

14.5 

7.5 

3 

9 

15 

20 

18 

17.5 

10 

7 

3 

4.5 

13 

11 

5.5 

9.5 

13 

16 

12 

14 

9.5 

6 

11 

13 

2 

1.5 

1 

1 

14 

12.5 

9.5 

10.5 

10 

15 

9.5 

14.5 

16.5 

12 

16 

18 

17.5 

16.5 

17.5 

17 

1 

1.5 

2 

2 

18 

9.5 

9.5 

7.5 

4.5 

19 

11 

6.5 

13 

7.5 

20 

4 

12.5 

4.5 

4.5 

REFERENCES  FOR  SUPPLEMENTARY  READING 

Rugg,  H.  O.     Statistical  Methods  Applied  to  Education,  pp.  134-147. 
Thorndike,    E.    L.     Mental    and    Social  Measurements,  pp.   51-59, 
Chapters  XII,  XIV. 

1  Adapted  from  M.  E.  Haggerty's  "Arithmetic:  A  Cooperative 
Study  in  Educational  Measurements,"  Indiana  University  Bulletin, 
Vol.  XII,  No.  18,  p.  443. 


CHAPTER   IX 

USELESSNESS    OF    STATISTICS   IN    CURRENT 
SCHOOL   REPORTS 

I.    THE    SITUATION 

Thoughtful  school  men  will  agree  with  Professor 
Hanus  in  his  statement  that  many  of  the  ordinary  presen- 
tations of  statistical  material  on  schools  to  the  public  are 
useless.1  In  his  investigation,  which  included  a  random 
selection  covering  the  entire  country,  he  found  that  the 
reports  studied  by  him  were  "  vague  in  purpose,  miscel- 
laneous in  subject  matter,  and  hence  ineffective."  2  He 
found  that  many  of  the  tables  were  printed  just  as  they 
had  been  tabulated  by  the  superintendent ;  that  exactly 
50  per  cent  of  the  reports  studied  had  no  adequate  inter- 
pretations of  tables ;  that  only  42  per  cent  contained 
any  comparative  statistics  whatever. 

1  Compare,  for  example,  the  statement  of  Superintendent  Giles 
of  Richmond,  Ind. :  "Mu.ch  of  the  labor  expended  on  school  reports 
is  lost  because  they  are  not  in  a  form  useful  for  comparison."  This 
is  contained  in  an  article  in  Educational  Administration  and  Super- 
vision, Vol.  II,  p.  305,  where  he  attempts  "to  make  available  for  ad- 
ministrative purposes  a  part  of  the  annual  reports  of  city  superin- 
tendents in  Indiana  to  the  State  Department  of  Public  Instruc- 
t  ion." 

2  Hanus,  Paul  H:  "Town  and  City  Reports"  fmore  particularly 
superintendents'  reports;,  School  and  Society,  Vol.  Ill,  p.  196 

200 


Statistics  in  Current  School  Reports      201 

H.    CAUSES    OF   THE   USELESSNESS 

Hanus  makes  the  point  that  "  the  useless  tables  of 
statistics  are  often  quite  useless  because  they  are  mere 
collections  of  working  data  unrelated  and  uninterpreted, 
and  also  because  they  pertain  only  to  the  year  under  re- 
view." l  Or  as  he  puts  it  in  another  place,  "  without  inter- 
pretation or  discussion,  many  statistics  are  meaningless 
to  most  readers,  whether  lay  or  professional."  2 

Hanus  believes  that  a  part  of  this  confusion  exists 
because  one  report  is  gotten  out  for  several  classes  of 
people  who  really  need  separate  reports.  He  says  that 
the  superintendents  must  give  information  concerning 
schools  to  three  classes :  (1)  the  school  board ;  (2)  the 
teaching  staff;  (3)  the  public.  The  so-called  annual 
report  is  usually  gotten  up  for  the  three  classes  with  one 
or  two  more  or  less  consciously  in  mind,  but  it  often 
fails  to  be  serviceable  to  any  of  the  three.  "  Reports  to 
the  board  or  staff  should  always  contain  both  the  statistical 
summaries  and  the  details  from  which  they  are  derived. 
.  .  .  Reports  to  the  people  should  contain  only  sum- 
maries,  by  items,  of  course,  and  for  successive  years  or 
periods ;  and  special  pains  should  be  taken  to  interpret 
them  so  clearly  that  he  who  runs  may  read."  3  The 
report  to  the  people  should  be  "an  abstract  or  digest  of 
the  reports  to  the  board,  brief  but  comprehensive  and 
clear."  In  addition  to  the  three  classes  noted  by  Hanus 
in  his  thoroughly  representative  reports,  some  super- 
intendents apparently  write  for  two  other  classes : 
(1)  the  state  or  national  educational  officials,  (2)  students 
of  education,  as  indicated  by  Snedden  and  Allen.4     But 


1  Ibid.,  p.  196  2  Ibid.,  p.  195  » Ibid.,  p.  196 

1  School  Reports  and  School  Efficiency,  pp.  4,  5 


202  School  Statistics  and  Publicity 

this,  of  course,  only  makes  the  reports  all  the  worse  for 
the  public. 

Whatever  the  relative  values  of  the  different  reports, 
the  superintendent  should  consider  very  carefully  the 
preparation  of  the  school  statistics  for  the  public.  If 
this  is  to  be  done  effectively,  he  must  first  master  the 
statistical  material  in  hand  and  get  hold  of  the  significant 
points  to  be  found  in  the  data.  For  help  on  this,  he 
may  consult  previous  chapters.  But  his  time  spent  in 
preparing  the  special  report  for  the  public  will  not  be 
lost  for  either  himself  or  his  teachers.  Honest  statistics 
gotten  up  to  influence  the  public  will  influence  school 
men  and  teachers  all  the  more  strongly  because  the 
striking  points. set  forth  so  vividly  in  the  popular  presen- 
tation will  make  more  impression  on  teachers.  The  old 
saying  that  we  never  completely  know  a  thing  until  we 
can  tell  it  effectively  to  some  one  else  applies  here.  The 
superintendent  never  fully  understands  his  own  statistics 
and  their  relations  until  he  can  present  the  significant 
points  effectively  to  the  public.  In  short,  to  paraphrase 
Snedden  and  Allen,  in  the  main,  the  methods  that  will 
give  the  maximum  of  publicity  on  school  statistics  will 
probably  result  also  in  providing  the  most  effective  statis- 
tical basis  for  school  administration.1 

III.    DEVICES   FOR   EFFECTIVE    PRESENTATION 

But  granted  that  the  superintendent  realizes  the  in- 
effectiveness of  his  school  statistics  and  is  determined 
to  issue  a  good  report  specifically  for  the  public,  what 
devices  are  available  for  him?  There  are  only  three 
main  devices  for  this  purpose.     He  may  graph  his  statis- 

1  School  Reports  and  School  Efficiency,  p.  8 


Statistics  in  Current  School  Reports     203 

tics;  he  may  translate  them  into  words;  or  he  may- 
tabulate  them.  Each  device  is  best  suited  for  certain 
conditions.     Each  has  its  strengths  and  weaknesses. 


1.    Graphic  Presentations 

A  graph  or  drawing  can  be  so  constructed  as  to  show 
at  a  glance  the  significant  things  in  a  mass  of  technical 
data.  Then,  too,  it  is  so  very  simple  that,  if  properly 
made  up,  it  can  be  readily  grasped  by  the  average  man 
who  is  totally  unversed  in  the  intricacies  of  statistical 
manipulation.  In  general,  however,  it  cannot  be 
accurately  remembered  or  reproduced  without  consider- 
able effort. 

2.    Translation 

By  translation  is  meant  the  transferring  or  inter- 
pretation into  ordinary  language  of  statistical  ideas, 
relationships,  and  results.  These  need  to  be  translated 
for  the  ordinary  man  before  they  can  be  clear  to  him, 
much  less  affect  him  to  any  appreciable  extent.  Such 
translation  is  difficult  because  sometimes  it  is  as  hard  to 
give  statistical  results  through  mere  words  as  it  is  to 
describe  a  painting  in  words  only.  But  the  superintendent 
in  employing  translation  has  only  the  same  problem  as 
has  any  able  editorial  writer,  any  successful  campaign 
orator,  any  writer  on  agricultural  or  food  topics,  or  any 
writer  for  a  society  engaged  in  propaganda  on  a  large 
scale.  Some  of  the  best  effects  produced  by  such  persons 
are  managed  through  translating  statistical  material  in 
word  language  only.  A  good  translation  can  be  easily 
remembered  and  used  in  a  speech  or  article  without  any 
special  preparation. 


204  School  Statistics  and  Publicity 

3.    Tabulation 

Probably  on  the  whole  the  least  serviceable  device  is 
tabulation.  The  ordinary  tabulations  and  masses  of 
statistics  in  most  school  reports  are  of  very  little  value 
for  the  public.  The  significant  facts  do  not  stand  out  in 
tables  of  this  sort.  It  requires  much  labor  for  any 
person  other  than  the  maker  of  an  ineffective  table  to 
get  anything  out  of  it.  Certainly  it  would  be  unreasonable 
to  expect  the  average  man  to  waste  any  time  on  tables 
of  this  nature,  even  if  he  can  be  persuaded  to  read  the 
rest  of  the  report.  In  such  tables,  comparisons  with 
similar  data  are  usually  impossible.  If  facts  about  a 
number  of  ward  schools  in  the  system  are  given,  the 
arrangement  may  be  alphabetical,  and  the  reader  cannot 
find  whether  School  A  is  better  than  School  B  without 
some  labor.  Norms  or  standards  with  which  any  item 
in  the  table  might  be  compared  may  be  left  out.  There 
may  be  no  units  of  measurement.  Often  great  masses 
of  facts  related  slightly  or  not  at  all  to  each  other  are 
thrown  together  in  the  same  table. 

Even  if  helpful  devices  are  employed  in  tabulation, 
or  if  the  table  is  arranged  like  a  scale  with  the  largest 
magnitude  at  the  top  and  constantly  decreasing,  or  if 
heavy  type  is  used  for  the  home  city  or  special  school,  or 
if  the  table  is  simplified  by  breaking  it  up  into  parts  and 
so  on,  still  the  graphs  and  translation  are  probably  best 
for  the  public.  In  either  of  the  latter,  the  significant 
facts  are  outstanding  and  may  be  grasped  by  the  reader 
with  less  labor  than  he  will  have  to  use  in  understanding 
even  good  tabulations.  The  impression  that  the  graphs 
and  translation  make  is  more  striking  and  lasting. 

However,  tabulations  are  necessary  in  the  statistical 


Statistics  in  Current  School  Reports      205 

process  because  they  must  precede  the  graph  or  trans- 
lation. The  graph  is  only  a  pictorial  representation  of 
the  significant  facts  in  a  tabulation.  The  translation  is 
only  a  word  representation  of  the  same  thing.  Often- 
times we  do  not  know  what  the  things  we  wish  to  graph 
or  translate  for  the  public  are,  until  we  have  made  the 
proper  tabulation.  Consequently  we  shall  take  up  in 
detail  the  matter  of  tabulating  school  statistics,  following 
it  with  treatments  of  graphs  and  translations. 


CHAPTER  X 

PRESENTING    TABULATED    STATISTICS    TO    THE 

PUBLIC 

I.    POSSIBILITIES   OF  USING   TABULATIONS   OF   STATISTICS 
TO    INFLUENCE   THE   PUBLIC 

As  previously  shown,  one  of  the  chief  reasons  for  the 
many  useless  tables  of  statistics  in  school  reports  is  that 
the  average  school  superintendent  tends  to  publish 
without  modification  the  tables  he  gets  up  for  his  own 
use  or  for  state  and  national  requirements.1  But  the 
mere  publishing  of  such  tables  is  about  the  worst  thing 
he  can  do  if  he  is  trying  to  influence  his  public.  It  is 
practically  certain  that  the  majority  of  people  who  read 
school  reports  at  all  tend  to  skip  conscientiously  most  of 
the  tabulated  matter  of  this  nature.2 

What,  then,  is  the  object  of  preparing  a  tabulation  for 

1  See  p.  201 

54  This  does  not  in  any  way  militate  against  the  contentions  of 
Professor  Thorndike  and  others  that  educational  investigators  should 
get  into  the  habit  of  publishing  in  full  the  data  secured  in  their  studies. 
Such  complete  publications  are  highly  desirable,  but  not  for  the 
general  public.  They  should  appear  in  educational  periodicals,  mono- 
graphs, dissertations,  and  similar  places.  If  included  in  reports  to 
the  public,  they  should  at  least  be  in  appendices  where  they  will  not 
prevent  the  citizen  from  reading  the  parts  of  the  report  that  will  in- 
fluence him.  But  even  thus  they  will  probably  weaken  the  effect 
of  the  other  parts  upon  him. 

206 


Presenting  Statistics  to  the  Public       207 

the  public?  The  original  intention  undoubtedly  is  to 
give  the  public  quickly  and  easily  a  bird's-eye  view  of  a 
great  mass  of  data  which  are  entirely  unintelligible 
until  collected  and  systematized.  Again,  tabulation 
achieves  a  great  economy  of  space  through  combining 
many  items  which  had  been  isolated  up  to  that  time. 
This  saving  of  space  will  not  be  achieved  without  some 
care.  For  example,  it  is  possible  to  make  numerous 
separate  tables  with  all  the  headings  repeated  for  each, 
as  in  reporting  facts  about  all  the  grades,  so  that  the 
total  occupies  more  pages  than  straight  reading  matter 
giving  the  same  facts  would  take  up.  But  in  general,  the 
argument  is  that,  if  the  ordinary  reader  can  understand 
the  tabulation  and  will  read  it,  he  is  more  likely  to  be 
influenced  by  it  than  by  the  pages  of  print  necessary  to 
give  the  same  idea.  There  is,  however,  a  great  danger  in 
undue  condensation,  especially  in  getting  too  many 
items  into  one  table.  For  example,  the  1914-15  report 
for  Des  Moines  has  one  table  with  ninety  different 
columns.1  The  ordinary  reader  can  hold  in  mind  only 
a  few  different  items  at  one  time.  It  is,  therefore,  out 
of  the  question  to  count  on  such  a  complicated  table's 
influencing  the  public.  And  it  is  probably  unwise  to 
insert  a  table  of  this  kind  in  a  report  intended  even 
partially  for  public  consumption.  A  third  object  in 
tabulation  is  the  aid  to  logical  thinking  and  presentation 
which  it  affords  for  all  work  capable  of  expression  in 
numbers.  Most  of  us  recall  how  our  science  teachers 
insisted  that  the  extent  to  which  we  mastered  scientific 
thinking  would  be  shown  by  our  laboratory  reports. 
Probably  these  reports  first  acquainted  us  with  tabulation. 

1  Annual  Report  of  the  Des  Moines  Public  Schools,  for  the  year 
ending  July  1,  1915,  opposite  p.  82 


208  School  Statistics  and  Publicity 

But  the  distinction  between  tabulations  intelligible  to 
school  men  or  experts  and  those  easily  understood  by  the 
public  cannot  be  overemphasized.  Tabulation,  although 
a  very  useful  device  for  work,  does  not  come  from 
instinct.  It  must  be  acquired,  a  truth  often  forgotten 
by  the  expert  tabulator.  Most  of  us  probably  recall 
some  good  mathematician  who  makes  a  very  poor  in- 
structor because  he  has  forgotten  how  painfully  slow  was 
his  own  mastery  of  some  of  the  work  he  is  teaching.  He 
has  consequently  no  patience  with  his  students.  Similarly 
a  superintendent  familiar  with  tabulation  is  very  apt  to 
forget  that  the  public  is  not  so  familiar.  It  is  true  that 
many  a  business  man  makes  much  use  of  tabulation  in 
his  work.  He  has  been  trained  to  do  this,  however,  for 
no  man  can  successfully  conduct  a  business  involving 
many  details  unless  he  knows  how  to  collect  and  con- 
solidate data  concerning  his  complex  enterprise,  in  what 
corresponds  to  tabular  form. 

But  even  an  expert,  trained  to  see  data  thrown  into 
certain  conventional  or  familiar  forms,  is  easily  upset 
if  confronted  with  a  tabulation  differing  from  the  form 
to  which  he  is  accustomed.  If  a  table  is  arranged  with 
magnitudes  running  from  low  to  high  instead  of  from 
high  to  low,  or  if  the  order  of  years  runs  from  the  most 
recent  to  the  most  remote  instead  of  vice  versa,  or  if 
any  similar  change  is  made,  the  expert  accustomed  to 
his  own  different  tabulations  will  often  hesitate  and 
become  lost  for  a  while.  It  may  take  him  some  time  to 
understand  the  new  forms.  The  ordinary  man  would 
not  give  even  that  much  time  to  trying  to  understand 
them.  He  would  pass  by  the  table  the  instant  that  he 
could  not  easily  grasp  its  meaning. 

It  should  be  evident  by  now  that  the  making  of  tab- 


Presenting  Statistics  to  the  Public        209 

illations  to  influence  the  public  is  no  easy  task.  It  is 
indeed  of  so  difficult  a  nature  as  to  require  study.  We 
shall  now  proceed  to  make  this  study,  taking  up  first  the 
problem  of  how  to  give  a  bird's-eye  view  through  tab- 
ulation, and  following  it  with  a  consideration  of  how  to 
get  up  a  series  of  tables.  The  reader  should  clearly 
understand  that  all  this  presupposes  an  acquaintance 
with  the  preliminary  treatment  of  blanks  and  tabulating, 
pages  71-81. 

II.    HOW    TO    GIVE    A    BIRD'S-EYE    VIEW    THROUGH 
TABULATION 

Many  points  are  to  be  considered  in  making  a  table  for 
the  public.  But  it  should  before  all  else  strive  to  give 
the  reader  quickly  a  bird's-eye  view  of  the  data.  Among 
the  things  required  for  this  are : 

1.    A  Very  Carefully  Worded  and  Specific  Heading 

The  heading  should  be  so  well  chosen  as  to  give  the 
key  to  the  understanding  of  the  table,  even  if  no  other 
explanatory  material  follows. 

As  examples  of  incomplete  headings,  the  following  from  the  South 
Bend  Survey  are  pertinent : 

(a)  Table  headed  "Value  of  School  Property"  (page  13). 

It  contains  the  dates  of  the  erections  of  the  various  build- 
ings, sizes  of  the  lots,  value  of  buildings  and  equipment. 

(b)  Table  headed  "Registration"  (page  15). 

It  contains  not  only  the  registration  by  years  but  the  in- 
crease in  registration  and  average  daily  attendance. 

(c)  ,Table  headed  "Cost  of  Fuel,  1912-13"  (page  21). 

This  is  by  ward  schools  and  contains  the  number  of  tons 
each  school  used,  the  total  cost  in  a  lump,  and  the  cost  per 
pupil. 

In  none  of  these  does  the  title  really  specify  what  is  in  the  table. 


210  School  Statistics  and  Publicity 

2.    Comparatively  Few  Classes  of  Facts  in  One1   Table 

Of  course,  it  is  hard  to  say  just  how  many  classes  of 
facts  should  go  into  one  table.  But  it  is  safe  to  say  that  a 
table  containing  more  than  four  such  classes  will  not  be 
read  by  the  average  man.  By  far  the  greater  portion  of 
the  tables  in  the  best  school  surveys  are  concerned  with 
only  one  class  of  facts,  and  those  including  as  many  as 
three  classes  are  very  rare.  No  one  will  question  the 
statement  that  the  best  methods  of  presenting  school 
statistics  to  the  public  have  so  far  appeared  in  these 
surveys  or  in  annual  reports  that  are  virtually  surveys. 

3.    Careful  Grouping  of  the  Classes  That  are  Used 

A  complete  running  list  of  items  is  of  little  value.  It 
corresponds  to  the  term  of  derision  used  toward  a  person 
who  puts  too  many  items  into  a  story  or  explanation, 
"  total  recall." 

Thus  in  the  Birmingham,  Alabama,  report  for  1914-15  there  is  a 
bird's-eye  tabulation  of  five  city  school  systems,  covering  twelve 
items  (pages  18  and  19).  In  the  Des  Moines  report  for  1914-15 
there  appears  a  table  of  ninety  columns  (grouped  by  sixes,  however), 
which  contains  the  judgments  of  ninety  judges  on  some  specimens 
of  handwriting  (page  82).  This  table  will  never  be  read  by  the  public. 
The  Birmingham  report  for  1914-15  contains  a  table  of  "Distribu- 
tion of  Expenditures"  (page  8)  which  has  twenty-six  columns  not 
broken  in  any  way.  Many  similar  examples  might  be  cited.  In  all 
probability  none  of  these  tables,  so  far  as  the  average  man  is  con- 
cerned, is  worth  printing. 

Once  in  a  while  a  long  table  of  receipts  and  expenditures 
is  justified  on  the  ground  that  the  citizens  demand  it. 
They  may  demand  it  even  though  they  will  never  read  it, 
because  they  think  it  will  secure  greater  honesty  and 
efficiency  in  handling  school  moneys.     But  such  a  table 


Presenting  Statistics  to  the  Public       211 

may  be  more  tolerable  by  using  double  entry  as  in  Table 
22. 

Table  22.     Double  Entry  Table  Showing  Receipts  and  Pay- 
ments, Springfield,  Illinois,  Public  Schools,  1912-13  * 


Receipts 

Balance  at  beginning  of  year      .     .     . 

— 

$17,414.19 

314,788.96 

4,176.84 

2,538.50 

2,807.40 

297.28 

264,597.87 

Total     

— 

606,621.04 

Payments 

Board  of  Education  office       .... 

Operation  of  office  building    .... 
Instruction 

Stationery  and  supplies 

Operation  of  plant 

Fuel 

Light  and  power 

Janitors'  supplies 

$2,353.68 

40.00 

720.00 

1,295.50 

730.00 

3,000.00 

1,000.00 

6,550.00 

30,177.25 

169,012.99 

325.00 

8,914.90 

3,807.25 

16,420.91 
6,519.37 
2,046.08 
1,268.05 
1,050.00 

— 

Springfield,  Illinois,  Survey,  p.  99 


212 


School  Statistics  and  Publicity 


Maintenance  of  plant 

Repair  of  buildings  and  care  of  grounds 

Repair  and  replacement  of  equipment  . 

Insurance . 

Auxiliary  agencies 

Libraries 

Promotion  of  health 

Miscellaneous 

Rent 

Permanent  outlays 

Land 

New  buildings 

Alteration  of  old  buildings     .... 

New  equipment  of  new  and  old  build- 
ings      

Other  payments 

Redemption  of  bonds 

Payments  of  interest 


$16,276.98 

5,346.11 

533.10 

850.00 
1,434.00 

120.00 

9,249.80 
85,527.91 
20,000.00 

2,963.61 

10,500.00 
6,663.75 


Total 414,696.24 


Balance  at  end  of  year 


Grand  Total 


191,924.* 


$606,621.04 


However,  although  Table  22  gives  the  classified  feature, 
it  does  not  appear  on  one  page  and  could  not  be  gotten  on 
one  page  in  print  large  enough  to  be  read  easily.  It  would 
make  a  better  showing  if  printed  on  adjacent  pages,  a  thing 
easily  possible  in  reports  but  not  feasible  here. 

Long  tables  are  also  broken  into  smaller  units  through 
the  use  of  subdivisions  of  the  main  classes  of  items.  The 
question,  however,  arises  as  to  how  far  such  subdivision 
may  extend  without  making  the  table  too  complicated. 
Certainly  the  process  of  subdivision  should  not  go  farther 
than  the  description  of  school  buildings  in  Table  23. 


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214 


School  Statistics  and  Publicity 


Long  tables  may  also  be  broken  up  by  grouping  the  items. 

Thus,  contrast  the  tables  of  Cincinnati  and  Cleveland  for  com- 
parison of  expenditures  for  different  years,  as  given  by  Snedden  and 
Allen,  pages  35-36,  here  shown  as  Tables  24  and  25 : 


Table  24.    Example  op  Effect  of  Unclassified  Items, 
parison  of  School  Expenditures  for  the  Years 
1895-1905  (Cincinnati) 


Com- 


Year  ending 

Year  ending 

Year  ending 

Aug.  31,  1895 

Aug.  31,  1900 

Aug.  31,  1905 

Teachers,  day  schools 

$669,752 

$799,286 

$815,719 

Teachers,  night  schools    . 

9,606 

6,612 

8,321 

Officers  and  examiners     . 

15,143 

16,646 

17,792 

Librarians 

New  buildings    .... 

Repairs 

Lots 

Furniture 

Heating  fixtures      .     .     . 

Fuel 

Printing     ...... 

Advertising 

Gas 

Census 

Textbooks     and     supple- 

mentary readers 

1,641 

3,502 

13,448 

Incidentals,  etc.       .     . 

5,678 

4,643 

3,284 

Teachers'  Institute       .     . 

Interest  and  redemption 

Public  library     .... 

Deaf-mute  taxes     .     .     . 

Transfer  of  funds    .     .     . 

Presenting  Statistics  to  the  Public       215 


Table  25.    Example   of    Effect  of   Classified   Items.     Com- 
parison of  School  Expenditures  (Cleveland) 


August  31 

1900 

1905 

Tuition 

Maintenance 

Officers'  and  employees'  salaries  .... 

Fuel  and  light 

$37,406 
897,190 

118,664 
1,295 

$50,964 
1,314,660 

184,144 

Repairs 

Water 

10,335 

Fixed  charges 

Interest    

Bonds 

Permanent  improvements 

Land 

Buildings 

Grading,  paving,  etc 

Improvement  on  existing  buildings  .     .     . 

Miscellaneous 

School  books 

Supplementary  reading  books 

Glenville  annexation 

( 
Total       

216  School  Statistics  and  Publicity 

4.  Arrangement  of  Totals  So  That  They  Can  Be  Quickly 

Grasped 

This  will  aid  the  superintendent  to  discover  significant 
relations,  as  well.  The  most  common  devices  are  the 
use  of  the  word  "  total,"  the  employment  of  black-face 
type  or  italics,  and  the  outset  in  another  right-hand 
column ;    thus : 

$  673  $   673  $  673 

1240  1240  1240 

890  890  890     $2803 

Total.     $  2803  $  2803 

5.  Printing  of  Headings  So  They  Can  All  Be  Read  from 

One  Way 

It  is  not  pleasant  to  have  to  twist  one's  head  or  turn 
the  book  around  to  read  vertical  headings.  The  average 
reader  will  skip  such  rather  than  do  the  turning.  For 
the  same  reason,  tables  that  are  printed  lengthwise  on 
the  page  should  be  avoided  if  at  all  possible.  In  general, 
the  headings  may,  by  the  use  of  syllables  and  abbre- 
viations, be  horizontally  printed  in  just  as  small  a  space 
as  the  wrong  way  takes.     See  pages  72,  213. 

6.  Filling   in   Tables    Only   Where   Data   Actually   Exist 

If  vacant  places  are  filled  with  zeros,  the  labor  involved 
in  reading  the  zeros  is  as  heavy  as  that  in  reading  the 
actual  numbers.1  However,  if  the  paper  is  not  cross- 
lined,  it  is  better  to  fill  the  vacant  spaces  with  dotted 
lines,  as  the  reader  may  get  lost  if  several  vacancies  occur 
together.     Compare    for   example    Tables    26    and    27 2. 

1  This  docs  not  in  any  way  discount  what  was  said  on  page  84  about 
making  an  entry  for  every  case.  That  referred  to  a  blank  for  the 
superintendent  to  use  in  collecting  data,  not  for  the  public. 

2  Sneddon  and  Allen  :  School  Reports  and  School  Efficiency,  p.  78 


Presenting  Statistics  to  the  Public       217 


Table  26.     Showing   Use   of   Zeros  —  Detroit   Central 
High  School 

Total  number  of  "  first  year  "  pupils  who  have  left  since  Sept.,  1904 


Cause 


Ages 


12 


B1  G    T 


13 


B    G    T 


14 


B    G    T 


15 


a 

(j 

1 

4 

0 

1 

0 

1 

10 

2 

2 

1 

0 

3 

2 

0 

0 

1 

4 

6 

Totals 


B    G     T 


Illness 

Illness  in  family 
Failing  eyesight 

Work 

Transferred  .  .  . 
Left  city  .... 
Indifference  to  work 

Music 

Unknown 

Total     .... 


1  1 

0  0 

0  0 

0  0 

0  0 

0  0 

0  2 

0  0 

0  0 


23 
4 
2 

41 
6 

10 
9 
1 

23 
119 


Table    27.     Omitting   Zeros  —  Detroit    Central    High 
School 

Total  number  of  "first  year"  pupils  who  have  left  since  Sept.,  1904 


Ages 

Totals 

Cause 

n 

13 

14 

15 

B;G 

T 

B 

G 

T 

B 

1 

1 
5 

G 
1 

1 

1 
1 

1 

T 

2 

1 
6 
1 

1 

1 

B|G 

1 

5 
1 
1 
12 
3 
3 
2 

1 
10 

B 

5 

1 
34 

4 
3 
8 

10 
65 

G 

18 
4 
1 
7 
2 
7 
1 
1 

13 
54 

T 

Illness 

Illness  in  family 

Failing  eyesight      .     .     . 

Work 

Transferred 

Left  city 

Indifference  to  work   . 

Music 

Unknown       

Total 

i 

1 

•) 

1 

1 

1 

10 

2 

9 
4 

4 
1 
1 
2 
1 
3 

1 
6 

23 
4 
2 

41 
6 

10 
9 
1 

23 
119 

1  In  order  to  include  all  the  columns  within  the  limits  of  the  page,  it  has 
been  necessary  to  use  the  abbreviation  B  for  Boys,  G  for  Girls,  and  T  for 
Total. 


218  School  Statistics  and  Publicity 

7.    Care  to  Avoid  Eyestrain 

All  the  type  in  a  table  should  be  large  enough  to  be 
read  easily  and  quickly.  It  is  a  false  economy  to  compress 
tables  into  a  very  small  space  through  the  use  of  fine  print. 
This  may  save  money  on  printing  bills.  But  it  loses 
more  money,  because  the  fine  print  will  not  be  read  by  the 
very  persons  for  whom  the  table  was  prepared. 

Dotted  lines  are  very  helpful  to  aid  the  eye  in  covering 
long  horizontal  stretches  between  data  that  are  to  be 
connected.  They  should,  however,  go  as  far  as  they  are 
needed. 

In  Table  28  dotted  lines  start  out,  but  do  not  go  far  enough  to  be 
of  real  service. 

Table  28.     Example  of  Bad  Use  of  Dotted  Lines.     Cost  for 
Overhead  Administrative  Control  in  Western  Cities  l 

City  Per  cent  of  total  mainte- 

nance cost  spent  for  ad- 
ministrative control 

Sacramento,  Cal 1.8 

Spokane,  Wash 2.2 

Pasadena,  Cal 2.4 

Seattle,  Wash 2.6 

Oakland,  Cal 2.7 

Denver,  Colo 2.7 

etc.  etc. 

8.    Avoidance  of  the  Alternate   Column  Scheme  Unless 
the  Types  and  Arrangement  in  the  Columns  Stand  Out 

Sharply 

For  example,  it  is  usually  unwise  to  use  alternate  lines 
for  statistics  on  boys  and  girls,  for  public  consumption, 
unless  one  set  is  in  red  type,  black-face  type,  or  italics. 
Italics  as  used  by  Ayres  in  his  "  Laggards  in  Our  Schools  " 

1  Oakland  Survey,  p.  17 


Presenting  Statistics  to  the  Public       219 

are  generally  less  satisfactory  than  black-face  type.  As 
red,  for  practical  purposes,  can  be  used  only  on  individual 
copies  and  not  in  printed  reports,  it  is  often  better  to 
make  a  separate  table  for  boys  and  one  for  girls  if  they 
are  to  be  printed.  This  simplifies  the  matter.  Both 
tables  may  be  later  summarized  into  one,  as  on  page  217, 
for  purposes  of  comparison  which  will  be  clear  to  any  one 
who  has  looked  at  the  previous  tables. 

In  the  Des  Moines  report  for  1914-15,  a  table  of  this  kind  appears, 
opposite  page  98.  It  is  an  age-progress  table.  Children  who  have 
always  been  in  school  in  Des  Moines  are  shown  in  the  upper  left-hand 
column  of  each  rectangle;  those  coming  in,  in  italics  in  the  upper 
right-hand  column  ;    and  the  total  below  in  heavy  type  : 


20     SU 
54 


The  differentiation  in  linos  within  divisions  of  a  table 
must  also  be  carefully  attended  to.  Note  in  Tables  26 
and  27,  page  217,  how  lighter  lines  are  used  to  subdivide 
the  larger  divisions.  Dotted  lines  would  have  done  as 
well  for  the  interior  lines,  but  the  use  of  colons  for  this 
purpose  is  not  advisable. 

The  Salt  Lake  City  Survey  has  a  table  x  in  which  colons  have  been 
used  to  indicate  interior  lines.  But  as  set  up,  they  appear  at  first 
to  indicate  ratios  between  boys  per  seat  and  the  percentage  of  suffi- 
ciency, and  so  on.  It  takes  the  average  reader  some  time  to  realize 
that  these  colons  are  really  intended  to  form  dotted  dividing  lines. 

9.    Clear  Headings  and  Subheadings 
i 
The  subheadings  must  be  clear  without  much  explana- 
tion.    The  use  of  only  key  numbers  or  letters  to  head 
columns  is  bad  enough  for  the  person  doing  the  work.2 

1  Page  249  2  See  page  74 


220  School  Statistics  and  Publicity 

Every  effort  should  be  made  to  avoid  it  in  a  tabulation 
intended  for  the. average  man. 

On  page  50  of  the  Hammond  Survey  appears  a  table  employing  key- 
letters  which  actually  occupy  almost  as  much  space  for  explanation 
as  the  tabulation  itself. 

10.    Neatness  and  Artistic  Features 

In  addition  to  the  foregoing,  it  is  highly  desirable  to 
have  all  printed  tables  as  neat  and  artistic  as  possible. 
Tables  that  are  pleasing  to  the  eye  will  by  their  very 
form  and  convenience  attract  unconscious  attention, 
which  may  then  easily  be  diverted  to  a  consideration  of 
the  ideas  or  conclusions  embodied  in  the  tabulated 
material.  Or  they  will  cause  the  reader  so  little  strain 
that  whatever  attention  he  has  to  give  them  will  be 
concentrated  on  the  thought  alone.  A  competent  printer 
will  be  able  to  secure  such  results  by  himself  but  may 
need  to  be  held  down  to  setting  up  all  type  so  that  it  can 
be  read  from  one  position.  Where  the  superintendent 
has  to  give  complete  directions  for  printing,  he  should  be 
especially  careful  about  uniform  headings,  uniform  spac- 
ing, large  print,  inclosing  each  table  in  a  border,  or  at 
least  using  distinctive  ruling  or  open  spaces  above  and 
below  each  table  to  set  it  off  properly.  The  tables  in 
this  book  have  been  planned  to  serve  as  models  on  these 
matters. 

III.    HOW    TO    MAKE    UP    A    SERIES    OF    TABLES    OF    THE 
SAME    GENERAL    NATURE 

Often  it  is  desired  to  have  a  series  of  tables  of  the 
same  general  nature  or  at  least  closely  related.  This 
necessitates  : 


Presenting  Statistics  to  the  Public       221 

1.  A  Summary  Table  at  the  Start  and  Minor  Tables  in 
the  Same  Sequence  as  Items  in  the  Summary  Table 

For  example,  if  it  is  desired  to  give  a  series  of  tables 
on  various  school  costs  per  pupil,  the  procedure  might 
be  thus :  First,  have  a  summary  table  with  various  items, 
as  cost  of  superintendent's  office,  cost  of  instruction, 
cost  of  supervision,  etc.,  all  appearing  in  the  same  table 
for  the  city  as  a  whole.  This  could  be  followed  by  a 
series  of  tables,  each  covering  all  the  ward  schools  on  one 
item.  These  tables  would  follow  the  sequence  of  the 
items  in  the  summary  table. 

2.  The  Same  Sequence  of  Items  within  Similar  Tables 

If,  in  these  various  tables,  the  lump  sum,  the  average 
number  of  pupils  belonging,  and  the  cost  per  pupil  are 
given  in  adjacent  columns  from  left  to  right,  it  is  advisable 
to  follow  the  same  order  of  presentation  and  the  same  form 
of  table  all  the  way  through.  In  this  way  the  "  mind 
set  "  of  the  reader  may  be  utilized. 

3.  Keeping   in   Mind   That   the    Main   Purpose   of   Any 

Tabulation  Is  the  Showing  of  Relationships 

(a)  The  purely  alphabetical  order  of  items  in  many  tables 
destroys  or  greatly  handicaps  the  showing  of  relationships. 

Thus  in  the  report  of  the  city  superintendent  of  South  Bend  for 
1913-14,  all  the  tables  involving  the  different  wards  or  schools  are 
practically  worthless  for  influencing  the  public  because  of  this  alpha- 
betical arrangement.  For  example,  take  the  table  on  comparative 
costs  of  instruction  and  supervision  by  buildings  for  1913-14,  page  30. 
If  the  last  column  of  this  table  had  been  arranged  in  order  of  magni- 
tude from  high  to  low,  the  comparison  would  stand  out.  See  Tables 
29  and  30  for  this. 

The  original  table  might  do  if  the  central  tendency  and  quartiles 
for  the  city,  or  some  other  standards,  were  printed  in  bold-faced  type 


222 


School  Statistics  and  Publicity 


at  the  top  of  the  table,  so  that  comparison  might  be  made  with  them 
for  each  ward  school.  Or  it  would  not  be  so  unsatisfactory  if  another 
column  had  been  added  at  the  right  giving  the  rank  of  each  school 
on  cost  per  pupil,  from  highest  to  lowest.  Grammar  would  be  rank 
2;  Colfax,  rank  9;  and  so  on  to  Warren,  which  would  be  rank  3. 
But  the  ranks  would  not  show  up  so  well  as  in  the  arrangement  on 
page  223.  In  any  event,  it  is  only  just  to  note  that  the  alphabetical 
arrangement  is  of  service  in  making  easy  the  work  of  checking  the 
names  of  the  schools  so  that  no  one  will  be  omitted.  This  has  its 
value  during  the  period  of  working  up  data  inside  the  school  system. 
But  this  value  disappears  as  soon  as  the  table  reaches  the  citizen.  He 
assumes  that  the  work  is  correct  and  desires  only  to  get  at  the  meaning 
of  the  whole  and  its  parts  as  quickly  as  possible. 

ORIGINAL   FORM 
Table  29.    Example  of  Alphabetical  Arrangement  of  Items 
Showing    Comparative    Cost   of   Instruction   and   Super- 
vision by  Buildings  for  1912-13  (South  Bend) 


Total 

Average 

Cost 

Schools 

cost  of 

daily 

per 

instruction 

attendance 

pupil 

Grammar 

$10,054.05 

332 

$30.28 

Colfax 

etc. 

etc. 

24.53 

Coquillard 

25.36 

Elder       

22.19 

Franklin 

21.04 

32.35 

22.26 

27.97 

20.14 

24.07 

Linden     

25.10 

Madison 

26.12 

27.94 

Oliver 

19.90 

Perlev 

23.61 

River  Park 

19.25 

Studebaker 

23.89 

Warren 

29.16 

Presenting  Statistics  to  the  Public       223 

REVISED   FORM 

Table  30.    Bobbitt  Table  Arrangement  of  Data  in 
Table  29 

Schools  Cost  per  pupil 

Jefferson $32.35 

Grammar 30.28 

Warren 29.16 

Lafayette 27.97 

Muessel 27.94 

Madison 26.12 

Coquillard 25.36  The  full  figures  from  which 

Linden 25.10       this    table    is    derived    are    on 

Colfax 24.53       file  in  the  superintendent's  office 

Lincoln 24.07       and  may  be  inspected  by  any 

Studebaker 23.89       interested  person. 

Perley        23.61 

Kaley 22.26 

Elder 22.19 

Franklin 21.04 

Laurel 20.14 

Oliver 19.90 

River  Park 19.25 

(6)  It  is  generally  best  to  arrange  the  table  in  the  form 
of  a  scale  running  from  high  to  low. 

There  are  some  exceptions.  Perhaps  a  better  rule 
would  be  to  arrange  items  in  the  table  so  that  the  city  or 
school  having  the  desired  trait  in  the  greatest  abundance 
should  be  at  the  top. 

For  example,  a  table  showing  the  number  of  dollars  behind  each  $1 
spent  for  schools  should  be  placed  with  the  smallest  number  at  the 
top.  The  smaller  number  is  the  more  desirable,  since  a  small  number 
of  dollars  behind  each  dollar  spent  for  schools  indicates  a  high  school 
tax  and  presumably  good  schools.  Accordingly,  Table  31  should 
have  been  reversed. 


224  School  Statistics  and  Publicity 

Table  31.     Real    Wealth    Behind    Each    Dollar    Spent    for 

School  Maintenance  l 

1.  Atlanta,  Ga $559.00 

2.  Los  Angeles,  Cal 538.00 

3.  Richmond,  Va 536.00 

4.  Birmingham,  Ala 479.00 

5.  Portland,  Ore 456.00 

6.  Memphis,  Tenn 449.00 

7.  Indianapolis,  Ind 408.00 

and  so  on  to 

35.  Toledo,  Ohio 184.00 

36.  Worcester,  Mass 180.00 

37.  Newark,  N.  J. 165.00 

Note  that  the  ranks  in  this  are  really  reversed.  Atlanta  in  this* 
showing  is  doing  less  for  schools  than  any  of  the  other  cities  and  should 
have  rank  37.     Newark  should  have  rank  1. 

Table  32,  which  gives  the  same  idea  in  another  way,  is  correct  in 
putting  the  largest  number  at  the  top,  because  a  high  tax  rate  on  real 
wealth  for  schools  is  desirable.     It  is  given  this  way : 

Table  32.     Comparative  Rates  op  Tax  Required  for  School 
Maintenance  (in  Mills)  2  Based  on  Real  Wealth  of  Cities  3 

1.  Newark,  N.  J 00606 

2.  Toledo,  Ohio 00543 

3.  New  Haven,  Conn 00541 

4.  Paterson,  N.  J 00541 

5.  Lowell,  Mass 00515 

and  so  on  to 

31.  St.  Paul,  Minn .     .00244 

32.  Memphis,  Tenn 00244 

33.  Portland,  Ore. 00219 

34.  Birmingham,  Ala 00209 

35.  Richmond,  Va 00186 

36.  Los  Angeles,  Cal 00184 

37.  Atlanta,  Ga 00180 

1  Portland  Survey,  p.  310 

2  In  some  parts  of  the  country,  this  would  be  more  easily  under- 
stood if  given  as  cents  on  the  one  hundred  dollars;   as : 

1.    Newark,  N.  J $.606 

3  Portland  Harvey,  p.  311 

i 


Presenting  Statistics  to  the  Public       225 

In  printing  Bobbitt  tables,  the  most  effective  results 
on  the  casual  reader  will  undoubtedly  be  obtained  by 
giving  each  item  a  separate  line. 

Thus  in  Tables  33-35  from  pages  62  and  63  of  the  San  Antonio 
Survey,  Form  1  is  best,  Form  2,  the  next  best.  Form  3  should  not  be 
used.  A  trained  reader  can  understand  one  form  as  easily  as  the 
other.  But  the  average  man  will  understand  the  first  form  much 
more  quickly  than  he  will  the  others. 


Table  33.    Bobbitt  Table,  Form  1.    Annual  Per  Capita 
Expenditures  for  Street  Maintenance,  1912 

Nashville $2.79 

Augusta 2.76 

Tampa 2.10 

Memphis 2.04 

Houston 2.04 

Savannah .  1.71 

Atlanta           '.     '.     '.     '.     '.     '.     '.     '.  1.63 

Dallas 1.55 

Galveston 1.54 

Jacksonville 1.53 

Austin        1.51 

New  Orleans 1.50 

MaconT     '.     '.     '.     !     '.     '.     '.     '.     '.  1.43 

Shreveport 1.36 

Montgomery 1.36 

Mobile '.  1.33 

Fort  Worth 1.17 

El  Paso 1.10 

f               Muskogee 1.07 

Birmingham 1.02 

San  Antonio .99 

Charleston .85 

Little  Rock    .......  .63 

Oklahoma  City  .                     .     .     .  .63 


226 


School  Statistics  and  Publicity 


Table  34.    Bobbitt  Table,  Form  2.    Annual  Per  Capita 
Expenditures  for  Street  Maintenance,  1912 

Nashville.     .....  $2.79 

Augusta 2.76 

Tampa 2.10 

Memphis 2.04 

Houston 2.04 

Savannah 1.71 

Atlanta 1.63 

Dallas 1.55 

Galveston 1.54 

Jacksonville 1.53 

Austin 1.51 

New  Orleans      ....  1.50 


Macon     .     .     . 

$1.43 

Shreveport  .     . 

1.36 

Montgomery    . 

1.36 

Mobile     .     .     . 

1.33 

Fort  Worth  .     . 

1.17 

El  Paso   .     .     . 

1.10 

Muskogee    .     . 

1.07 

Birmingham 

1.02 

San  Antonio 

.99 

Charleston   .     . 

.85 

Little  Rock  .     . 

.63 

Oklahoma  City 

.63 

Table  35.  Bobbitt  Table,  Form  3  (Practically  Never  De- 
sirable). Annual  Per  Capita  Expenditures  for  Street 
Maintenance,  1912 


Nashville $2.79 

Tampa 2.10 

Houston 2.04 


Augusta $2.76 

Memphis 2.04 

Savannah 1.71 


Atlanta 1.63 

Galveston      .     ...     .     .  1.54 

Austin 1.51 

Macon 1.43 

Montgomery      .     .     .     .  1.36 

Fort  Worth 1.17 


Muskogee 1.07 

San  Antonio        ....         .99 
Little  Rock 63 


Dallas 1.55 

Jacksonville      ....  1.53 

New  Orleans     .     .     .     .  1.50 

Shreveport 1.36 

Mobile 1.33 

FA  Paso_.     ._._  .  _^     ! 1.10 

Birmingham      .     .     .     .  1.02 

Charleston .85 

Oklahoma  City      .     .     .  .63 


The  use  of  Forms  2  and  3  probably  grows  out  of  a  desire  to  utilize 
part  of  a  page  for  the  table,  or  through  a  mistaken  idea  that  it  econ- 
omizes space.  But  in  most  places  where  it  is  desired  to  use  such  a 
table,  there  would  be  many  such  tables  to  present.  It  is  readily 
apparent  that  two  tables  of  Form  1  placed  side  by  side  on  a  page 
would  take  up  practically  no  more  space  than  if  they  were  printed 
either  as  Form  2  or  Form  3. 


Presenting  Statistics  to  the  Public       227 

(c)  If  many  horizontal  lines  appear  in  the  table,  ease 
in  reading  them  is  facilitated  by  running  a  heavy  horizontal 
line  or  leaving  a  space  every  five  lines  or  less. 

Examples  of  this  are  familiar  to  most  readers.  The  lines  drawn 
across,  or  the  gaps  left  in  the  Bobbitt  tables  to  show  the  medians  and 
quartiles  serve  the  same  purpose,  if  there  are  not  too  many  cases  in 
each  section  of  the  distribution.  A  third  device  for  this  purpose  is 
the  numbering  of  the  items  from  top  to  bottom  on  both  margins. 
Thus  the  data  for  item  3  on  the  left  may  be  traced  across  to  the  data 
on  the  line  numbered  3  on  the  right.  This  is  the  device  used  by  the 
United  States  Bureau  of  Education  on  many  of  the  tables  that  cover 
two  pages,  with  continuous  horizontal  lines. 

(d)  In  most  tables  where  the  cases  are  kept  separate, 
it  is  advisable  to  make  the  name  of  the  city,  school,  etc., 
to  be  compared  with  the  others,  stand  out  prominently. 

It  will  be  recalled  that  this  was  done  in  the  tables  from  the  Port- 
land Survey.  The  device  is  the  familiar  one  used  by  newspapers  to 
call  attention  to  the  home  city  in  a  table  giving  the  standing  of  the 
baseball  clubs  of  a  league.  Sometimes  the  emphasis  is  given  with 
capital  letters  as  in  Portland,  or  with  black-faced  type  as  in  Salt  Lake 
City.  On  paper  charts  not  to  be  printed,  the  particular  case  can  be 
marked  in  red  or  some  bright  color.  It  is  hardly  possible  to  make  the 
one  case  and  the  data  with  it  stand  out  too  prominently. 

In  the  age-grade  tables,  it  is  customary  to  make  the  children  of 
normal  age  stand  out  prominently  by  marking  off  these  numbers. 
Heavy  stairstep  lines  may  be  drawn  to  inclose  the  numbers  for  nor- 
mal children ;  or  heavy  lines  may  be  placed  above  and  below  these 
numbers ;  or  they  may  be  boxed  in ;  or  they  may  be  printed  in  bold- 
faced type.     Samples  are  given  in  Figure  29. 

Where  the  printer  can  handle  it,  the  boxed-in  form  is  preferable, 
because  it  enables  one  to  separate  the  normal  from  both  the  retarded 
and  the  accelerated  children  very  easily. 

In  'another  form  of  age-grade  table,  the  facts  for  one  grade  at  a 
time  are  presented.  This  is  usually  done  by  running  the  ages  along 
the  top  and  the  number  of  years  the  child  has  been  in  school  down  the 
table.  Then  two  heavy  lines  are  drawn  vertically  through  the  table 
to  inclose  the  children  of  normal  age  in  this  grade.     Two  other  heavy 


228 


School  Statistics  and  Publicity 


Agas 


Grades 

I  B 
I  A 
IB 
JLA 
HTB 
3EA 


under 
6 

6 

to 

6* 

to 

7 

7 

to 

7* 

7* 

to 

8 

22 

54-0 

159 

105 

38 

30 

98 

"118 

76 

17 

65 

133 

102 

2 

27 

5S 

84- 

Ag 

es 

Grades 

under 
6 

6 

to 

6t 

t6* 

to 

7 

7 

to 

7* 

7* 

fo 

8 

1  B 

22 

54-0 

159 

105 

38 

IA 

30 

98 

118 

76 

hb 

17 

65 

133 

102 

DA 

2 

27 

58 

84 

uib 

HA 

Ages 


Fig.  29. 


under 
6 

6 

to 

to 

7 

7 

to 

7i 

7i 

to 
s 

22 

540 

»59 

105 

38 

30 

98  J    118 

76 

17 

65 

133 

102 

2. 

27 

58 

B4 

Grades 
IB 
I  A 

IB 

ha 

UIB 
JffA 


Devices   for   Making   the   Number   of   Normal  Children  Stand 
Out  in  Agu-CJrade  Tables. 


Presenting  Statistics  to  the  Public       229 

lines  are  drawn  horizontally  through  to  inclose  the  children  of  normal 
progress  irrespective  of  how  old  they  were  at  entrance.  The  area 
in  the  center,  where  the  two  sets  of  lines  cross,  incloses  children  who 
are  both  normal  for  age  and  progress.  Various  other  combinations 
for  the  other  parts  of  the  table  are  easily  worked  out. 

The  following  table  form  is  similar  to  one  used  in  the  Bridgeport, 
Connecticut,  Survey,  page  35. 


Table  36.    Age-Progress  Table.     5B  Grade 


Years 


in 
School. 

Ages 

Total. 

8 

H 

9 

3k 

10 

io| 

ii 

I'Z 

12 

I2i 

13 

2 

2* 

3 

3k 

4 

+2 

5 

5Z 

6 

6i 

7 

Total' 

(e)  It  is  often  advisable  to  present  only  percentage 
derivations  instead  of  the  original  figures. 


230  School  Statistics  and  Publicity 

The  reason  is  that  the  original  figures  may  be  large  and  not  easily 
comparable,  whereas  the  percentage  equivalents  are  small  and  easily 
compared.  Distribution  tables  of  the  results  on  standard  tests,  how- 
ever valuable  for  educational  investigations,  are  not  at  all  suited  for 
the  general  public.  Round  numbers  or  the  nearest  whole  units  are  often 
better  than  exact  figures,  as  they  can  be  held  in  mind  more  easily. 
If  all  the  parts  of  100  per  cent  are  shown,  care  should  be  taken  to  as- 
sign values  to  the  parts  that  will  total  exactly  100  per  cent.  Approxi- 
mations are  satisfactory  for  the  superintendent,  but  a  discrepancy  be- 
tween the  sum  of  the  parts  and  100  per  cent  would  afford  some  readers 
an  excuse  for  attacking  the  accuracy  of  the  report.  These  suggestions 
must,  however,  be  used  with  caution.  For  in  some  instances  the  use 
of  approximate  figures  may  lead  readers  to  suspect  that  the  figures 
have  been  "  doctored."  Especially  is  this  true  as  regards  statistical 
presentations  designed  to  increase  the  school  tax. 

(/)  Special  devices  in  tabulating  are  sometimes  of  great 
value. 

For  example,  a  tabulation  showing  the  part  of  each  dollar  of  school 
money  that  goes  to  various  school  expenses  is  effective  with  the  average 
man.  Insurance  companies  take  great  pains  to  show  where  every  part 
of  the  dollar  from  premiums  goes.  Tables  37-39  show  some  of  the 
possibilities.  Similar  devices  could  show  the  part  of  a  year  or  years 
spent  on  each  subject. 

Table  37.    How  Portland  Spends  Its  Dollar  ' 

Interest 20.7  cents 

General  expenses  of  city  government        6.0  cents 

Police  department 9.0  cents 

Fire  department 12.0  cents 

Inspection 0.9  cents 

Health 0.7  cents 

Street  cleaning  and  sanitation 6.2  cents 

Care  of  streets  and  bridges 9.0  cents 

Education 30.8  cents 

Libraries 1.0  cents 

Parks  and  playgrounds •    .     .  2.6  cents 

Damages 1.1  cents 

Total 100.0  cents 

1  Adapted  from  the  Portland  Survey  Graph,  p.  84 


Presenting  Statistics  to  the  Public       231 

Table  38.    How  the  Trolley  Nickel  is  Divided  * 

2.02  cents  for  wages 

.78  cents  for  expenses 

.55  cents  for  taxes 
1.06  cents  for  interest 

.59  cents  for  dividend 
5.00  cents  —  your  nickel 

Table  39.    How  Rockford  Spent  Its  School  Dollar,  1915 2 

Elementary  Schools 

Teachers'  salaries 41.67  cents 

Building  maintenance  and  upkeep    .     .     12.3 

New  buildings       9.9 

Educational  supplies 3.9 

Department  of  hygiene 97  68.74  cents 

General 

Interest  on  school  fund 2.81 

Executive  employees  (Educational)  .  1.39 
Executive  employees  (Board)  ...  .86 
Evening  schools  and  gymnasium      .     .         .4  5.46 

High  School 

Teachers'  salaries 15.81 

Upkeep  of  building 6.7 

Educational  supplies 1.43 

New  building 1.41 

Educational  employees 45  25.80 

100.00  cents 

Sometimes  the  variations  in  sizes  of  different  items  may  be  in- 
dicated by  difference  in  sizes  of  the  type  used  in  printing.  Table  40 
is  a  good  example  of  this,  but  the  variations  in  size  are  only  very 
rough   approximations. 

1  Poster  used  by  Louisville,  Ky.,  Railway  Company 

2  Adapted  from  the  chart  on  p.  120  of  the  Review  of  Rockford  Public 
Schools,  1915-16.  Some  of  the  figures  were  changed  slightly  so  as  to 
make  the  total  exactly  100  cents. 


232 


School  Statistics  and  Publicity 


Table  40.    Variations  in  Type  to  Indicate  Relative  Size 

"Watch  the  Central  Association  Grow!" 


Year 


New  members  added    Total. paid  up 
during  year  membership 


1908 
1909 

1910 

1911 
1912 
1913 

1914 
1915 

1916 

1917 


102 
161 

I32 
117 

115 
146 
177 

223 

320 

1  Every  member 
get  one !  " 


324 
445 


484 
486 
559 
619 

680 
768 

973 

Are  you  one 
of  these  ? 


Net  increase 

121 

39 

2 

73 
60 

61 
88 

205 

"  There  is  no 
going  back." 


Do  you  need  more  convincing  proof  of  the  worth  of  membership  in 

The  Central  Association  of  SCIENCE  and  MATHEMATICS  TEACHERSP 

Table  41  uses  very  little  space,  for  data  classified  in  several  ways. 

Table  41.  Example  of  Presenting  in  a  Small  Space  Data 
Classified  in  Several  Ways.  The  Standing  of  Salt  Lake 
City  in  the  Fundamentals  of  Arithmetic  as  Compared  with 
Other  Cities,  Judged  by  the  Median  Score  Attained  by 
Each  Grade  1 


Addition 

Multiplication 

V 

VI 

VII 

VIII 

V 

VI 

VII 

VIII 

3.9 

4.6 

5.4 

6.7 

Detroit 

3.8 

4.8 

6.0 

7.5 

3.7 

4.9 

5.6 

7.8 

Boston 

3.3 

4.8 

5.1 

6.5 

3.9 

4.4 

4.7 

5.6 

Other  Cities 

2.6 

4.5 

5.2 

6.4 

2.9 

3.4 

3.8 

5.3 

Butte 

4.1 

5.0 

6.5 

8.1 

4.1 

6.4 

6.9 

8.5 

Salt  Lake  City 

4.3 

5.3 

7.1 

8.3 

Subtraction 

Dt 

'vision 

5.5 

6.2 

7.3 

9.5 

Detroit 

2.7 

4.4 

7.1 

8.8 

4.9 

6.3 

6.9 

8.6 

Boston 

2.0 

3.3 

5.1 

6.9 

4.5 

6.1 

7.8 

8.4 

Other  Cities 

2.3 

4.3 

5.8 

6.3 

2.9 

3.4 

3.8 

5.3 

Butte 

3.6 

4.3 

7.2 

10.2 

5.2 

7.8 

8.8 

9.8 

Salt  Lake  City 

3.0 

5.5 

7.7 

9.5 

1  Sal!  Lake  City  Survey,  p.  174 


Presenting  Statistics  to  the  Public       233 

(g)  There  are  times  when  it  is  necessary  to  use  con- 
ventional tables.  In  so  doing,  the  following  points  may 
be  of  use : 

1.  Use  double  distribution  tables  and  forms  recommended  by  the 
National  Education  Association  committee.1 

2.  Place  dates  old  to  new  down  the  page  or  left  to  right. 

3.  Place  magnitude  so  that  the  most  desirable  showing  is  at  the 
top  or  to  the  left.  (This  sometimes  reverses  the  order  of  dates  ad- 
vocated in  2.) 

4.  Use  Roman  numerals  for  one  classification,  e.g.,  the  numbers 
for  the  school  grades,  and  Arabic  numerals  for  the  data  on  pupils 
within  the  grades. 

5.  It  is  best  where  the  sense  is  not  destroyed  and  the  effect  de- 
sired will  not  be  lost,  to  get  up  tables  in  the  conventional  forms. 

Bizarre  effects  in  tabulation  are  no  more  to  be  desired  than  is 
writing  from  right  to  left  or  up  ,he  page. 

EXERCISE 

Take  some  annual  school  report  or  school  survey  in  which  you 
are  interested.  Write  out  a  detailed  criticism  of  the  tables  or  lack  of 
them  in  it,  from  the  standpoint  of  their  effectiveness  with  the  public, 
showing  just  why  they  are  good  or  liable  to  be  unsuccessful.  In  the 
cases  of  the  unsuccessful  ones,  or  failure  to  employ  tabulations  where 
desirable,  draw  up  forms  that  would  present  the  same  data  properly. 

REFERENCES  FOR  SUPPLEMENTARY  READING 

Report  of  the  Committee  on  Uniform  Records  and  Reports.     U.  S. 

Bureau  of  Education  Bulletin,  1912,  No.  3. 
Rugg,  H.  O.     Statistical  Methods  Applied  to  Education,  Chapter  X. 
Snedden,  David  S.,  and  Allen,  William  H.     School  Reports  and  School 

Efficiency. 

1  Report  of  Committee  on  Uniform  Records  and  Reports,  Bureau 
of  Education  Bulletin,  No.  2,  1912,  p.  20 


CHAPTER   XI 

GRAPHIC  PRESENTATIONS  OF  SCHOOL  STATIS- 
TICS,   ESPECIALLY   FOR   THE   PUBLIC 

I.    OBJECT   OF    GRAPHIC    PRESENTATIONS 

The  object  in  making  a  graphic  presentation  of  statis- 
tical matter  is  to  give  as  quickly  as  possible  through  the 
eye  a  faithful-  and  forceful  bird's-eye  view  of  the  mass  of 
statistics,  the  significant  parts,  their  relationships,  etc. 
Graphic  presentation  in  statistics  is  simply  a  develop- 
ment of  a  growing  general  tendency  to  make  desired 
impressions  by  pictures  rather  than  by  word  descriptions. 
This  tendency  is  shown  most  clearly  in  the  pictorial 
supplements  and  cartoons,  and  in  the  constantly  in- 
creasing proportion  of  illustrations  in  printed  material 
appearing  in  all  our  leading  newspapers  and  magazines 
at  the  present  time. 

The  graph,  however,  is  more  closely  akin  to  the  line 
drawing  than  it  is  to  the  photograph.  It  is  found  to  be 
of  great  advantage  in  textbooks  for  the  following  reasons : 

1.    It  presents  the  significant  points  in  a  clear  and  unmistakable 

These  points  are  also  presented  apart  from  the  great  mass 
of  subsidiary  data  on  which  they  rest. 
It  makes  the  presentation  concrete  by  appealing  to  the  eye. 

As   many   people   are   unable   to    understand    things    they 
cannot  image,  the  graph  will  drive  the  significant  points  home 
to  those  who  could  not  be  reached  otherwise. 
234 


way 


Graphic  Presentations  of  School  Statistics    235 

3.  It  often  economizes  time  and  space,  for  it  will  take  up  less  room 
than  the  description  it  displaces. 

4.  It  gives  to  most  persons  a  more  accurate  basis  of  comparison 
than  they  could  get  with  the  same  effort  from  word  descriptions  or 
tabulations. 

Textbooks  must  make  things  clear  very  quickly  to 
beginners  in  the  subject.  School  reports  must  present 
school  facts  very  rapidly  and  clearly  to  citizens  who 
know  little  of  them.  Consequently,  the  graph,  if  prop- 
erly used,  should  be  about  as  valuable  for  school  reports 
as  it  has  been  for  textbooks. 

The  graph,  however,  is  open  to  dangers  of  mis- 
representation and  exaggeration.  These  things  are  just 
as  harmful  in  school  graphs  as  they  are  in  demagogic 
politics,  patent  medicine  advertisements,  etc.  The  ideal 
graph  would  probably  have  the  forcefulness  of  the  most 
powerful  motor  car,  department  store,  or  patent  medicine 
advertisements,  with  something  like  the  truthfulness 
and  accuracy  on  essential  points  of  a  first-rate  scientist. 
Another  trouble  is  that  a  graph  leaves  with  most  persons 
only  a  general  impression ;  it  is  very  difficult  for  them  to 
recall  it  accurately,  much  less  to  reproduce  it  from  memory 
later  for  any  one  else. 

II.    HOW     TO     MAKE     GRAPHS     FOR    THE     PUBLIC     FROM 
STATISTICAL    DATA 

1.    Component  Parts 

Circle  Graph.  It  is  often  desirable  to  show  in  a  graph 
the  relative  size  of  the  component  parts  of  a  statistical 
whole.  A  popular  device  for  doing  this  is  the  circle, 
each  sector  by  its  area  representing  one  proportional  part 
of  the  whole. 


236 


School  Statistics  and  Publicity 


Figure  30  is  a  graph  of  this  nature  showing  the  distribution  of  time 
through  the  eight  grades  for  the  various  common  school  subjects. 
Figure  31  gives  a  similar  graph  with  fewer  parts. 


*e 


"(I, 


'Or 


'** 


•tr 


^ 


*ft' 


English,  12.86  7. 


Fro.  30.  —  Component  Part  Circle  Graph  Showing  Distribution  of  Time 
tf)  Various  Subjects  Throughout  the  Eight  Elementary  Craves. 

(Adapted  from  the  1915-10  Review  of  the  Rockford,  Illinois,  Schools,  page  52.) 

This  kind  of  graph  is  familiar  to  the  public  and  of 
course  makes  the  comparisons  through  the  sizes  of  the 
angles.     But  it  has  these  disadvantages  : 

1.  If  more  than  a  few  parts  are  to  be  represented,  then?  is  trouble 
in  reading  the  names. 


Graphic  Presentations  of  School  Statistics    237 


When  there  are  only  three  parts  and  they  are  all  large,  the 
names  may  all  be  printed  horizontally  as  in  Figure  31.  But 
if  there  are  many  parts  and  some  of  them  are  small,  this  is  im- 
possible. Then  the  names 
must  be  printed  as  in  Figure 
30  or  else  at  the  side  with 
dotted  lines  leading  into  the 
parts  indicated.  If  the 
former  device  is  used,  the 
printing  must  be  reversed 
as  the  eye  proceeds  around 
the  circle,  with  consequent 
delay.  Note  in  Figure  30 
how  one  has  to  read  up 
for  "music"  and  down  for 
"geography."  If  the  latter 
device  is  used,  much  more 
time  is  consumed  in  asso- 
ciating the  names  with  the 
parts  concerned,  than  when 
the  names  are  printed  on 
the  parts. 


Fig.  31.  —  Component  Part  Circle 
Graph     Showing     Relative     Propor- 
tions    of     Normal,     Retarded,     and 
Accelerated  Pupils. 
(From  Salt  Lake  City  Survey,  page  190.) 


2.  The  figures  denoting  the  various  parts  cannot  be  placed  in  such 
positions  that  they  can  be  easily  compared  or  added.  This  is  espe- 
cially bad  when  the  parts  are  given  in  percentages. 

3.  It  is  extremely  difficult  to  compare  the  same  factors  in  different 
wholes. 

For  example,  suppose  a  circle  similar  to  this  one  for  Rockford 
had  been  drawn  for  Joliet ;  it  would  be  very  hard  to  com- 
pare the  sector  on  "reading"  in  the  Rockford  circle  with  the 
corresponding  sector  in  the  Joliet  circle. 

Bar  Graph.  The  data  shown  in  this  circle  graph  for 
Rockford  may  be  presented  in  a  bar  graph  which  permits 
of  placing  the  figures  so  they  can  be  added.  For  this 
the  parts  within  each  of  the  three  main  divisions  should 
be  arranged  from  high  to  low.  Figure  32  shows  this 
arrangement. 


238 


School  Statistics  and  Publicity 


Note  that  practically  all  the  defects  of  the  circle  graph  have  been 
remedied  here.  Note  also  that  the  component  parts  in  a  bar  graph 
are  shown  proportionately  by  their  lengths  only.     Their  widths  have 

nothing  to  do  with  it  and  their 
areas  are  in  exactly  the  same 
proportion  as  their  lengths. 

Note,  too,  that  the  figures  are 
given  for  any  reader  who  cares 
for  them  and  in  such  shape  that 
they  may  be  added  easily. 

Sometimes  for  economy  in 
printing,  the  bar  may  be  a  hori- 
zontal one,  in  which  case  the 
lines  of  printing  and  figures  may 
run  vertically  so  that  the  reader 
will  have  to  turn  the  page  to 
read  them. 

An  elaboration  of  the 
component  bar  graph  is 
the  form  with  two  different 
but  related  scales,  one  on 
either  side. 

For  example,  the  waste 
through  repeaters  in  a  school 
system  might  be  shown  as  in 
Figure  33,  using  the  average 
annual  cost  per  elementary  pupil, 
say  $30. 

The  bar  graph  idea  with 
component  parts  may  be 
carried  out  for  popular 
presentation  with  cartoon 
effects. 

For  example,  the  proportion  of 
white  and  negro  children  out  of 


Reading 

20.72  % 

5.32  % 
14.75% 

6.38  % 
12.86% 

7.64  % 

2.01  % 
3.09  % 

5.58  % 
1.85% 

6.65  % 

3.26% 
1.09  % 
1.74  % 

7.06  % 

Penmanship 

Arithmetic) 

Spelling 
English 

Geography 

Physiology 
History 

Music 

M  T  and  D  5c 

Drawing 

Ph.Tr.  and  Rest 

Op.  Exercises 
Recess 

100.00  % 

Fig.  32.  —  Bar  Graph  Showing  by 
Component  Parts  with  Subdivisions 
the  Distribution  of  Time  for  Com- 
mon School  Subjects. 

(Adapted  from  Figure  30.) 


Graphic  Presentations  of  School  Statistics    239 


No.  of 
rspaaters 


50 
25 
50 

75 

50 
50 

50 

75 
425 


8th  Grad€ 

7th 

ii 

6  th 

•i 

5th 

■I 

4th 

a 

3rd 

•I 

2nd    " 

1st 

i 

Cost  of  repeat- 
ing  work 

#1500 

750 
1500 

2200 

1500 
1500 

1500 
2200 


12750 


Fig.  'S.i.  —  Bar  Graph  to  Show  Component  Parts,  with  Two  Different  but 
Related  >Scales. 

each  twelve  may  be  shown  as  in  Figure  34.  The  black  children  com- 
ing in  from  the  right  on  the  row  serve  the  same  purpose  as  putting 
the  right  end  of  a  bar  in  black.     But  the  idea  of  the  children's  figures 


Fig.  34.  —  Bar  Graph  with  Cartoon  Effect  Showing  Proportion  of   White 
and  Colored  Children  out  of  Every  Twelve  in  Alabama. 

(From  An  Educational  Survey  of  Three  Counties  in  Alabama,  page  IS.) 


240  School  Statistics  and  Publicity 

here  gives  an  interesting  touch  in  addition.  While  this  cartoon 
graph  is  not  accurate,  strictly  speaking,  a  little  care  will  make  it 
accurate  enough  for  its  purpose.  For  example,  by  choosing  twelve 
rather  than  ten  figures  for  the  whole  line,  fractions  of  two-thirds 
and  one-third  have  been  shown  with  whole  figures.  This  device 
is  extremely  effective  in  a  popular  presentation. 

2.    Simple  Comparisons 

The  simplest  graphic  comparison  is  usually  made 
through  some  form  of  the  bar  graph.  The  latter  is  used 
a  great  deal  for  comparisons  in  presentations  intended 
for  the  public,  but  its  possibilities  are  not  generally 
recognized.  Many  graphic  comparisons  for  public  con- 
sumption so  far  have  involved  one  or  both  of  two 
errors : 

(a)  The  comparison  is  made  by  using  similar  areas. 

(b)  The  comparison  is  made  through  a  cartoon  effect,  when  the 
latter  is  needed  only  to  attract  attention  to  the  graph,  in  which  bars 
would  show  the  relationship  much  better. 

We  shall  now  examine  the  possibilities  of  the  bar 
graph  for  comparisons,  and  after  that  take  up  typical 
graphs  of  other  kinds  for  this  purpose. 

Comparisons  with  Bar  Graphs.  Some  of  the  possi- 
bilities of  the  bar  graph  in  making  comparisons  have 
been  indicated  heretofore,  particularly  in  connection 
with  a  distribution  arranged  as  a  Bobbitt  table.1  If  the 
component  parts  of  any  bar  graph  on  pages  238  and  239 
were  taken  out,  arranged  in  order  of  size  and  lined  up  at 
the  left,  they  would  give  a  bar  graph  effect  running  from 
high  to  low. 

In  making  comparisons  with  bars,  it  is  generally  advis- 

1  See  p.  103 


Graphic  Presentations  of  School  Statistics    241 


able  to  follow  this  order,  with  the  bars  lined  up  at  the  left 
ends,  as  in  Figure  35. 

Name  of  item        Size  Bar 


Size 
$703 
5.52 


Newark 
Jersey  City 

Fig.   35.  —  Illustration  of  Correct  Order  of  Items  in  a  Bar  Graph. 


By  doing  this  the  magnitudes  will  come  in  a  column  where 
they  can  be  seen  by  those  who  like  to  have  the  figures, 
and  these  can  easily  be  added,  with  the  sum  at  the  bottom 
of  the  middle  column.  Very  seldom  should  the  numbers 
be  placed  at  the  immediate  right-hand  ends  of  the  bars. 
They  tend  to  make  the  bars  seem  disproportionately 
larger  and  they  cannot  be  easily  compared  or  added. 


lr  Vermont 
etcy  to 
46VLouisfana 


2o     30      40     50     eo      7o     so     so     too 


II 


mm 


Fig.  36.  —  Device  Used  by  Dr.  L.  P.  Ayres  to  Show  by  States  Percentages 
of  the  School  Population  Enrolled  in  Public  Schools,  in  Private  Schools, 
and  Not  in  Any  School,  in  1910. 

White  portion  indicates  children  in  public  schools  ;  shaded,  those  in  private  schools  ; 
and  black,  those  not  in  any  school.  (Adapted  from  A  Comparative  Study  of  the  Public 
School  Systems  in  the  Forty-eight  Stales  and  reproduced  by  permission  of  the  Russell 
Sa^e  Foundation.) 

To  make  the  bars  stand  out  clearly,  the  distance  between 
them  must  be  markedly  different  from  the  width  of  a 
bar.  ^Otherwise  the  lines  will  tend  to  run  together  or 
sink  into  the  background. 

Sometimes,  to  economize  space,  some  of  the  largest 
items  may  be  shown  with  double  bars.  A  familiar  example 
is  the  graph  used  by  the  American  Book  Company  on  its 


242 


School  Statistics  and  Publicity 


calendar.  On  this  a  long  double  bar  represents  the 
expenditures  in  liquor  and  a  short  single  bar,  the  expendi- 
tures in  textbooks.  This  is  very  doubtful  practice,  as  it 
greatly  lessens  the  difference  between  the  magnitudes  for 
the  untrained  reader. 

The  horizontal  bar  graph  is  also  very  valuable  where 
it  is  desirable  to  make  comparisons  between  component 
parts  in  similar  wholes.  For  this  purpose,  the  wholes 
are  represented  by  bars  of  the  same  width  and  length. 


Republican      \V/A      Progressive  KW,')         Democrat     r_ 

40  SO  60  70 


80 


0  10  20  30 

Alabama     mawkkwv 

to 


New  York 


Fig.   37.  —  Device  to  Show  Parts  of  a  Total  and  Also  to  Indicate  Relative 
Sizes  of  the  Totals. 

Adapted  from  Professor  Irving  Fisher's  chart  showing  parts  of  the  total  vote  for 
president  in  1912,  at  the  same  time  indicating  the  relative  voting  strength  of  each  state. 

Then  the  component  parts  for  the  same  item  are  lined  up  on  the 
left  margin,  and  those  for  another  item  will  appear  lined  up  on  the 
right  margin.     A  fine  example  is  shown  in  Figure  36. 

It  is  to  be  noted  that  this  will  not  work  well  for  more  than  two 
component  parts.  Observe  how  difficult  it  is  to  get  an  idea  of  the 
size  of  the  middle  item  representing  the  number  of  children  enrolled 
in  private  schools  when  this  item  for  different  states  is  compared. 

If,  for  any  reason,  it  is  desirable  to  make  comparisons 
between  the  relative  sizes  of  the  wholes,  the  widths  of 
the  bar  may  be  adjusted  accordingly. 

Mr.  W.  C.  Brinton  borrows  such  a  device  for  giving  an 
analysis  of  the  total  vote  for  president  in  1912  in  the 
forty-eight  states  from  Professor  Irving  Fisher.1     Each 

'  Brinton,  W.  C. :   Graphic  Methods  of  Presenting  Facta,  p.  10 


Graphic  Presentations  of  School  Statistics    243 

state  is  represented  by  the  same  length,  and  the  percentage 
given  to  each  party  varies,  while  the  number  of  voters 
also  varies  in  the  states.     (See  Figure  37.) 

The  following  are  some  of  the  places  in  school  work 
where  a  graph  of  this  kind  would  be  helpful : 


1.  It  could  be  used  in  the  Ayres  graph  given  before  for  comparing 
the  total  number  of  school  children  in  the  states. 

2.  In  a  chart  comparing  the  number  of  retarded,  normal,  and  ac- 
celerated children  in  a  number  of  cities,  the  relative  totals  of  children 
enrolled  could  be  shown  by  the  width  of  the  bars. 

100  \00 


/, 


SZ 


68 


55 


35 

38 

i 

13 

1 

19 

i 

y 

28 

12  12 

? 

3     ** 
2LJ  VA    I   Y/A     I   X/A     I    1//I     I    YA     I  ^T~l 
13'      14        15  16  17  18         19 

Fig.  38.  —  Bar  Graph  for  Comparing  Two  Things  Whoso  Proportions  Are 

Constantly  Varying. 

Columns  represent  number  of  hoys  and  girla  among  each  hundred  beginners  who 
remain  in  school  at  each  age  from  i:j  to  li).  Shaded  columns  represent  boys  and  white 
columns  girls.  (From  Spriny field  Survey,  page  52,  by  permission  of  the  Russell  Sago 
Foundation.) 


244 


School  Statistics  and  Publicity 


3.  If  the  distribution  of  the  cost  of  educating  one  child  were  being 
shown  for  various  items  such  as  instruction,  supplies,  etc.,  for  several 
cities,  the  width  of  the  bar  might  represent  the  total  amount  spent 
by  each  city  to  educate  one  of  its  children,  etc. 

The  bar  graph  may  be  used  to  compare  two  or  more 
things  whose  proportions  are  constantly  varying,  by  using 
a  different  shading  for  each  separate  kind  of  item. 

Thus  Dr.  Ayres  uses  Figure  38  to  show  the  number  of  beginners 
that  remain  in  school  at  various  ages  at  Springfield,  Illinois,  using 
shading  for  boys  and  white  for  girls.  The  reader  not  only  gets  a 
comparison  between  the  number  of  boys  and  girls  remaining  in  school 
at  any  given  age,  but  also  a  comparison  between  the  different  age  groups. 

Other  examples  are : 

1.  Enrollment  of  pupils  in  different  grades,  white  columns  for  boys 
and  black  columns  for  girls  (Cleveland  Summary  Volume,  page  85). 

2.  Contrast  in  percentage  of  retardation  of  white  and  colored 
pupils  by  grades,  shaded  bars  for  white,  black  bars  for  colored  pupils, 
all  on  horizontal  basis  (Louisville  Report  for  1911+  15,  page  27). 


School 
year 


Daily 
cost 


High 
schools 


>alanes 


I  WASHINGTON 

12  VLRMONT 

24  KANSAS        Y//// 

36  MARYLAND   [ 

48  ALABAMA 

Fig.  39.  —  Device  for  Showing  Relative  Standing  of  Several  Cases  on 
Each  of  Several  Items. 
Adapted  from  Ayres's  graph  showing  the  standing  of  the  forty-eignt  states,  by  per- 
mission of  tin-  Russell  Sage  Foundation.  The  highest  quarter  is  represented  by  white, 
the  second  quarter  by  light  shading,  the  third  quarter  by  dark  shading,  and  the  lowest 
quarter  by  black. 

An  elaborate  yet  easily  understood  chart,  making  com- 
parisons for  the  forty-eight  states  on  ten  different  items, 
is  found  in  Ayres's  bulletin  on  the  state  school  systems.1 

1  Page  32 


Ch  to  CM 

m  m  't 
co 1  — :  ro 


245 


246  School  Statistics  and  Publicity 

This  is  made  up  by  starting  at  the  top  with  the  best  state,  Washing- 
ton, and  going  down  to  the  lowest,  Alabama.  Each  state  is  repre- 
sented by  a  separate  bar,  and  each  bar  is  divided  into  ten  parts. 
Each  of  these  ten  parts  represents  an  item  on  which  the  state  is  graded. 
For  each  item,  the  states  in  the  lowest  quarter  have  that  part  black ; 
the  states  in  the  next  quarter  have  dark  shading;  the  states  in  the 
next  to  the  highest  quarter  have  light  shading ;  and  the  states  in  the 
highest  quarter  have  white.     (See  Figure  39.) 

This  form  of  chart  is  excellent  for  graphing  the  numerous  items  of  a 
summary  table.  The  use  of  the  different  forms  of  shading  uniformly 
through  the  table  enables  the  reader  quickly  to  locate  any  state's 
rank  on  any  item.  The  use  of  the  darker  shades  for  the  worst  ranks 
on  any  item  is  also  a  good  device,  for  it  would  have  a  tendency  to 
sting  the  pride  of  the  average  citizen.  Note  also  that  an  idea  of  the 
general  standing  of  any  state  on  the  ten  items  can  be  gotten  quickly. 
For  example,  Washington  appears  with  practically  white  spaces  and 
so  is  manifestly  very  high  on  the  whole.  Alabama  has  a  black  space 
on  every  item  and  so  is  very  low.     These  things  can  be  seen  at  a  glance. 

Such  a  graph  would  be  of  use  in  comparing  the  per- 
formances of  several  pupils,  of  several  teachers,  of  several 
classes,  of  several  rooms,  or  of  several  schools  where  each 
individual  case  had  been  ranked  on  a  number  of  qualities 
or  achievements. 

For  example,  the  showing  on  standard  tests  for  several  eighth 
grades  might  be  made  thus,  using  a  horizontal  bar  for  each  grade,  and 
a  vertical  column  for  each  test  used. 

The  Boston  Survey  has  a  graph  to  show  how  Boston's 
expenditures  compare  with  those  of  the  average  city, 
using  component  parts  of  a  bar  graph  that  is  a  square. 
It  is  reproduced  here  as  Figure  40,  but  experiments  show 
that  while  it  is  not  readily  understood  at  first  by  the 
average  school  man,  it  is  exceedingly  effective  once  it  is 
grasped. 

Sometimes  it  is  desirable  to  compare  two  distributions 
that  have  been  grouped  by  similar  steps,  for  the  typical 

i 


Graphic  Presentations  of  School  Statistics    247 


amount  of  the  same  quality  in  each  step.  In  this  case, 
a  set  of  bar  graphs  running  to  the  right  may  be  used  for 
one  distribution,  abutting  on  a  similar  set  of  bars  running 

HIGH  SCHOOL  EDUCATION.  PAYS 
YEARLY    INCOME 


HIGH      SCHOOL 
TRAINING 


NO  H.  S. 

TRAINING 


IN  HIGH  SCHOOL 
IN  HIGH  SCHOOL 
*  500 
750 
1,000 
I  150 
I,  550 


*  200 


#  7,337 


TOTAL 


*  5,112 


H.  SCHOOL    TRAINED   BOYS-WAGES  $3.50  PER  DAY 
NO    H.  SCHOOL    TRAINING -WAGES*  I  5'0  PER  DAY 
Fig.  41.  —  Example  of  Right  and  Left  Device  for  Comparing  Distributions 
with  Bar  Graphs. 
It  shows  average  yearly  ineome  of  high  school  graduates  as  compared  with  that  of 
persons  fiot  having  high  school  training.      (From  one  of  the  1917  folders  of  the  Agri- 
cultural Extension  Division  of  the  International  Harvester  Company.) 

to  the  left  for  the  other  distribution.  It  is,  however, 
difficult  to  compare  magnitudes  on  the  right  with  those 
on  the  left. 


248 


School  Statistics  and  Publicity 


An  example  is  the  graph  used  by  the  Bureau  of  Edu- 
cation, shown  in  Figure  41. 

From  this,  it  is  only  a  short  step  to  the  graph  which 
exhibits  the  facts  about  several  qualities  in  a  city  or 


70 


6.0 


5.0         4.0 


DECREASES      INCREASES 
3.0         2.0  1.0  1.0 


lstGrade 

ond 
3rd 

4th 

6 


th 


,7th 
8th 

9th 


10 


th 


Fig.  42.  —  Example  of  Right  and  Loft  Device  for  Comparing  Distributions 
with  Bar  Graphs. 

It  shows  changes  in  distribution  of  enrollments  by  grades  in  Baltimore  between  180!' 
and  HUM).      (From  Baltimore  Survey,  page  98.) 


school,  by  showing  increases  on  bars  extending  to  the1 
right  of  a  vertical  line,  and  decreases  by  bars  extending 
to  the  left  of  it. 

A  good  example  is  Figure  42  from  the  Baltimore  Survey,  showing 
the  change  in  enrollments  in  the  various  grades  by  percentages.     The 


Graphic  Presentations  of  School  Statistics    249 

decrease  in  the  first  grade  was  due  to  special  efforts  to  move  up  re- 
tarded children. 

This  form  is  preferable  to  that  in  Figure  41  because  the  bars  to 
be  compared  are  closer  to  each  other. 

The  "  Monument  "  Graph.  Areas  are  not  at  all  easy 
to  compare.  Such  comparisons  would  be  far  more 
effective  if  made  by  bar  graphs  lined  up  at  one  end  so 
that  the  comparison  would  merely  be  a  matter  of  com- 
paring the  lengths  of  the  various  bars. 

For  example,  take  the  "monument"  graph  which  is  frequently 
found  in  school  reports.  A  good  illustration  is  found  in  one  repre- 
senting the  number  of  pupils  enrolled  in  each  grade  in  the  Alabama 
Survey  of  Three  Counties,  shown  in  Figure  43. 

Here  each  stone  represents  the  enrollment  for  one  grade,  beginning 
with  the  first  grade  for  5423  pupils  and  topping  off  with  a  little  stone 
for  the  60  pupils  in  the  last  year  of  the  high  school. 

The  errors  in  using  this  sort  of  graph  are  as  follows : 
The  areas  of  the  "  stones  "  may  be  taken  into  account 
for  the  relationship  by  the  reader,  when  it  really  is  shown 
by  their  lengths  only.  As  there  is  no  common  point 
from  which  to  measure  either  the  length  or  area  of  the 
stones,  no  adequate  idea  of  relative  sizes  can  be  obtained 
from  such  a  graph.  The  effect  of  this  form  of  comparison 
is  to  make  the  difference  between  the  larger  and  the 
smaller  numbers  seem  less  than  it  is.  This  may  be 
shown  by  rearranging  the  data  in  the  graph  in  a  regular 
bar  graph  lined  up  at  the  left.  (See  Figure  44.)  Notice 
how  much  accentuated  the  differences  appear.  It  may 
be  contended  that  the  monument  graph  is  only  one  form 
of  the  graph  given  in  Figures  41  and  42.  That  is,  it 
really  has  two  halves  formed  by  an  imaginary  line  down 
the  center,  either  half  of  which  gives  a  simple  bar  graph 
effect.     But  if  that  is  the  case,  why  not  use  the  simple 


250  School  Statistics  and  Publicity 

bar  graph  alone  in  the  first  place?     When  children  are 
lined  up  in  "  stair  steps  "  to  get  their  varying  heights, 


A 


IV  H.S.  60 
IEH.S.  119 
H  H.S.   198 
||   I  H.S.  455 
|    7th  Gr.  915  | 


6th  6*1485 


5thGr:  1800 


4*6*2080 


S^Gr.  2132 


Z^Or.  2587 


Ist  Gr.  5423 


Fig.  43.  —  Monument    Graph    Showing    Number    of    Pupils  Enrolled    in 

Each  Grade. 
(From  An  Educational  Survey  of  Three  Counties  in  Alabama,  page  63.) 

they  are  all  placed  on  the  floor.  No  one  would  ever  think 
of  putting  their  waist  lines  on  the  same  level  and  then 
of  taking  account  only  of  variations  above  the  waist. 


0     nr    H.S. 


60 


4rh    grade       2060 


1st   qrade      -5423 


Eio.  44.  —  Graph  Showing  Apparent   Size  of  Certain   "Stones"  from   the 
Monument  Graph  Lined  up  at  One  Side. 

Comparisons  with  Circle  Graphs.  Errors  frequently 
arise  in  making  comparisons  with  circles.  It  makes  all 
the  difference  whether  the  comparison  is  made  through 
the  diameters  or  through  their  areas.  The  ordinary 
reader  tends  to  make  the  comparison  on  the  diameter 
basis.  If  the  circles  are  drawn  on  the  area  basis,  how- 
ever, it  is  apparent  that  the  comparison  will  not  be  so 
striking  as  the  maker  of  the  graph  intended.     If,  on  the 


Graphic  Presentations  of  School  Statistics    251 

other  hand,  circles  are  drawn  on  the  diameter  basis,  some 
readers  will  tend  to  overestimate  the  facts. 

The  graph  used  on  page  91  of  the  Cleveland  Survey,  Summary 
Volume,  is  a  good  example  of  the  difficulty  in  using  circles  on  the  area 


© 

© 

© 

Under  age  and 
rapid  progress 

Normal  age  and 
rap\d  progress 

Over  age   and 
rapid  progress 

(         3°         ) 

0 

© 

Under  age  and 
normal  progress 

Normal  age  and 
normal  progress 

Over  age  and 
normal  progress 

© 

o 

0 

Under  age  and 
slow    progress 

Normal  age  and 
slow     progress 

Over  age  and 
slow  progress 

Fig.  45. —  Comparison  of  Circles  by  Areas  Using  the  Percentage  of 
Children  in  Each  Age  and  Progress  Group  in  Elementary  Schools  of 
Cleveland  at  Close  of  Year  1914-15. 

(From  Cleveland  Survey,  Summary  Volume,  page  91.  By  permission  of  the  Survey 
Committee  of  the  Cleveland  Foundation.) 

basis.  It  is  reproduced  in  Figure  45.  Here  nine  circles  are  employed 
to  show  the  percentages  of  children  in  the  under-age,  normal-age,  and 
over-age  subdivisions  of  the  rapid,  normal,  and  slow  groups.      This 


252 


School  Statistics  and  Publicity 


requires  nine  circles  and  it  is  almost  impossible  to  compare  them  ac- 
curately. One  circle  is  marked  6  and  another  30,  but  the  former 
appears  to  be  about  one-fourth  of  the  latter,  or  larger  than  in  reality. 


A    Larger    Proportion   of  Children  are   Going  to    High   School. 

Fm.   4fi.  —  Concentric  Circle  Graph    to    Show    Relative    Increase    in    High 

School   Knrollnient. 

(From  tin'  1015-10  Review  of  tin   Hoc/; font,  Illinois,  Schools,  pape  107.) 


The  use  of  concentric  circles  as  a  means  of  comparison 
is  even  worse  than  that  of  the  circles  apart  from  each 
other. 


Graphic  Presentations  of  School  Statistics    253 

Figure  46  is  a  concentric  circle  graph  showing  the  enrollment  of 
the  Rockford  schools  for  the  years  1895,  1900,  1905,  1910,  and 
1915,  with  a  comparison  of  the  high  school  enrollment  with  the  total 
enrollment.  This  graph  is  worth  little  for  the  public,  because  it  is 
hard  to  understand,  and  because  it  is  almost  impossible  to  get  a  cor- 
rect notion  of  the  areas  of  the  different  circles.  One  tends  to  look 
only  at  the  rings  and  not  at  the  circles,  and  the  whole  effect  is  some- 
thing like  the  advertisements  showing  the  various  layers  in  an  auto- 
mobile tire. 

The  bar  graph  would  be  very  much  better  for  this 
comparison,  as  in  Figure  47. 

Comparisons  made  by  using  the  areas  of  segments  of 
the  same  circle  are  questionable,  because  people  have  not 

High  School  Grades 

•895    -    514.  niMlI  1 


to 
1915      -    6332 


Fig.  47.  —  Component  Bar  Graph  Comparison  of  Data  Shown  in  Figure  46. 

been  trained  to  estimate  the  areas  of  segments.  Even 
as  simple  a  graph  as  "  How  Portland  Spends  Its  Dollar  " 
(see  Figure  48)  is  hard  to  interpret  correctly.  But 
when  the  graph  becomes  as  complicated  as  the  one 
in  Figure  49,  it  is  probably  useless  for  the  average 
reader. 

It  is  even  more  difficult  to  compare  sectors  in  different 
circles  than  in  the  same  circle. 

For  example,  take  the  graph  on  the  percentage  of  home-trained  and 
non-trained  teachers  and  principals  in  the  Cleveland  Survey,  shown  in 
Figure  50.  The  sectors  here  serve  but  little  to  emphasize  the  different 
percentages  given.  In  addition,  the  white  labels  on  the  black  parts 
probably  cut  down  the  apparent  size  of  the  black  parts  very  mate- 
rially. 


254 


School  Statistics  and  Publicity 


Triangle  Graphs.  A  few  surveys  make  comparisons 
by  the  heights  or  areas  of  overlapping  isosceles  triangles 
having  equal  bases,  as  in  Figures  51  and  52. 


Interest 

20. 7+ 


'ParkslP-U! 


.0" 


£ 


& 


Department  \t± 


Fir;.  4<S.  —  Example   of    Difficulty  of    Comparing    Component  Parts   in   a 
Circle  Graph  When  the  Angles  Are  Not  Clearly  Shewn. 

This  figure  gives  an  itemized  statement  of  "How  Portland  Spends  Its  Dollar." 
(From  Portland  Surety,  page  84.) 

Since  two  triangles  with  equal  bases  are  to  each  other  as 
their  altitudes,  the  comparison  is  perfectly  accurate  from 
either  a  height  or  area  standpoint.  But  as  heights  alone 
are  really  wanted  for  determining  the  areas,  plain  bars 


"Diagram  III.  Surface  of  circle  represents  total  per  capita  expenditure  in  the 
average  city.  Sectors  are  proportional  to  amount  spent  for  each  of  the  twelve  main 
purposes  for  which  funds  are  expended.  Shaded  portion  represents  expenditure  in 
Bridgeport.  Under  each  heading  the  first  figure  gives  in  dollars  and  cents  the 
amount  .spent  per  child  per  year  in  the  average  city  and  the  second  figure  the  corre- 
sponding amount  for  Bridgeport." 


255 


256 


School  Statistics  and  Publicity 


Fig.  50.  —  Graph  Showing 
Difficulty  in  Comparing  Sec- 
tors from  Different  Circles. 

("From  Cleveland  Survey,  Sum- 
mary Volume,  pane  107.  By  per- 
rni-.-iori  of  tin'  Survey  Committee 
of  the  Cleveland  Foundation.) 


would  show  the  relative  lengths 
of  the  altitudes  with  much  less 
work.  Besides,  since  the  aver- 
age reader  has  had  no  experience 
in  comparing  heights  or  areas  of 
triangles,  about  the  only  justifi- 
cation for  them  is  the  one  matter 
of  adding  variety. 

Comparisons  with  Cartoon 
Effects.  Another  bad  use  of 
comparison  through  areas  is 
sometimes  found  in  the  employ- 
ment of  cartoons  in  which  the 
data  are  represented  by  the  areas 
of  persons  or  objects.  For  ex- 
ample, take  Figure  53.  The 
trouble  with  such  a  chart  is 
that  the  area  grows  much  faster 
than  the  height,  so  that  the 
expenditure  for  1914-15,  instead 
of  appearing  only  about  35  per 
cent  greater  than  for  1910-11  as 
it  should,  really  appears  to  be 
several  hundred  per  cent  greater. 
In  this  particular  chart,  the  hori- 
zontal lines  at  the  back  help  to 
reduce  the  exaggeration  by  em- 
phasizing the  height  factor. 

Even  when  the  figures  are 
given  with  the  chart,  as  in  this 
instance,  the  visual  inaccuracy 
is  serious  enough  to  cause  a  dis- 
trust of  the  whole  thing.     How- 


Graphic  Presentations  of  School  Statistics    257 

ever,  the  cartoon  effect  can  be  secured  in  a  graph  that, 
by  use  of  units  or  separate  figures,  allows  no  chance  for 
error  in  making  comparisons. 

A  very  effective  chart  of  this  kind  is  the  one  from  the  Des  Moines 
Report  for  1914-15.1  It  is  too  large  to  be  shown  effectively  in  this 
book,  so  will  be  described  in  words  only.  It  aims  to  show  in  a  car- 
toon the  relative  numbers  of  retarded  children,  normal  children,  and 


SALARIES      OP 

PRINCIPALS 


Fig.  51.  —  Comparison  by  Triangles 
between  the  Salaries  of  Principals  in 
Springfield  and  the  Average  for  Ten 
Other  Cities  in  1911-12. 

Shaded  triangle  represents  average  annual 
per  capita  expense  for  principals'  salaries  for 
each  child  in  average  attendance  in  the  day 
schools  of  Springfield,  and  triangle  in  outline 
represents  corresponding  expenditures  for 
the  average  of  ten  other  cities.  (From 
Springfield  Survey,  page  98,  by  permission  of 
the  Russell  Sage  Foundation.) 


WAGES      OF 
JANITORS 
*70 

455 


Fig.  52.  —  Comparison  by 
Triangles  of  the  Cost  of  Janitor 
Service  in  the  Average  City 
and  in  Bridgeport. 

Triangle  in  outline  represents 
portion  of  each  thousand  dollars 
spent  for  janitors'  wages  in  the 
average  city  ;  shaded  triangle  repre- 
sents the  amount  spent  in  Bridge- 
port. (From  Bridgeport  Survey, 
page  27.) 


accelerated  children,  boys  and  girls  separately.  To  the  left,  27  boys 
and  20  girls  are  represented  as  climbing  a  hill,  book  in  hand  and  read- 
ing. In  the  middle,  18  boys  and  23  girls  are  walking  along  on  a  level 
with  books  tucked  away  under  their  arms.  At  the  right,  5  boys  and 
7  girls, are  going  down  hill  with  no  books  at  all.  This  is  similar  to 
the  Alabama  illustration  in  that  it  gives  relative  proportions,  but  note 
how  much  better  this  relationship  is  shown  by  separate  children  than 
it  would  be  by  a  few  children  varying  in  size.  The  graph  would  have 
been  a  little  more  effective  if  each  group  of  children  had  been  in  one 

1  Page  99 


258  School  Statistics  and  Publicity 

line  so  that  the  length  of  the  line  might  also  have  entered  into  the 
comparison.  This  could  have  been  easily  managed  by  adding  a  few 
more  lines  to  indicate  a  portion  of  a  hill  for  each  child  not  on  the  level. 

See   how  the  child  in  the  grades  is  growing 
1910-11  1911-12        1912-13         1913-14         1914-15 


I=fc 


•::t  ::"*..  I 


$23^         $25^       $27^        $29^       $32^ 

Fig.   53.  —  Example  of  the  Difficulty  in  Making  Comparisons  with  Cartoon 

Effects. 
This  shows  what  is  spent  in  Louisville  on  each  child  in  the  grades.     The  comparison 
is  really  made  by  heights  only,  but  the  reader  tends  to  take  it  by  areas.    (From  1914-15 
Louisville  Report,  page  35.) 

This  would  take  no  more  room  on  the  whole,  and  it  would  not  seriously 
weaken  the  effect,  since  the  hill  conveys  practically  the  same  idea  as 
the  use  of  the  books. 

Another  example  of  a  cartoon  applying  a  good  idea  but  in  a  very 
unreal  way,  and  also  using  circles  for  comparison,  is  given  in  Figure  54. 

for  Every  dollar    that 

average  Bridgeport 

spenoh  spends 

BOARD  OF  EDUCATION  OFFICE  I  ■  *!  ^^     22t 


In;.  •">!.  —Cartoon  Graph   Using  a  Sector  of  a  Circle  to  Represent  Part 

of  a  Dollar. 

(From  Bridgeport  Survey,  page  22.) 

But  thp  cartoon  effect  could  have  been  kept  and  a  much  more  accurate 
comparison  made  by  using  cents  in  a  bar  effect,  thus : 

Average  city      $1.00 to  100 

Bridgeport  .22     .     .     .     to  22 


Graphic  Presentations  of  School  Statistics    259 

Ayres  uses  such  a  device  to  show  the  cost  of  schooling  per  child  per 
day  in  the  various  states,  in  his  bulletin  on  comparing  the  school 
systems  of  forty-eight  states.1 

When  the  cartoon  effect  is  very  necessary  to  show 
relative  parts  of  a  dollar,  and  accuracy  is  not  essential,  a 
cartoon  similar  to  Figure  55  may  be  advisable. 

ADVENTURES      OF      MR.     TAXPAYER. 


With     a     municipal       budget       and      without. 


How  one  city  now  How  a  city 

SUPPOSES  KNOWS 

the   money    is  spent.  where  the  mone_y  (joes. 

Fig.  55.  —  Cartoon  Effect  to  Show  Parts  of  a  Dollar. 
(From  Newburgh  Survey,  page  93,  by  permission  of  the  Russell  Sage  Foundation.) 

Figure  56  is  a  cartoon  effect  to  show  the  lack  of  suffi- 
cient playground  space. 

Figure  57  is  a  skillful  use  of  a  bar  graph  effect. 

Other  good  examples  of  cartoon  effects  may  be  found  in  : 

a.   The  Ohio  Survey,  page  65. 

Here  there  is  a  line  of  twelve  teachers  for  each  class  of  school,  begin- 
ning with  one-room  township  schools  and  going  on  up  to  high  school 
Teachers  without  professional  training  are  in  black ;  those  with  one 
or  more  terms  in  summer  schools  are  in  gray;   those  with  one  or 

1  Page  18 


260 


School  Statistics  and  Publicity 


more  years  in  a  professional  school  in  white.  A  high  degree  of  ac- 
curacy is  attained  by  using  half  a  figure  to  represent  one  twenty- 
fourth.  Thus  seven  twenty-fourths  of  the  twelve  teachers  in  one 
line  are  represented  by  three  women  clothed  in  black  and  another 
with  a  black  skirt  and  gray  waist. 

LAWN    vs.    PLAYGROUND 
William  Street. 


How  One 
Newburg 
School  Saves 
the  Grass  at 
the  Expense 
of  the 
Children 


PLAYGROUND 
BOYS     ONLY. 


Fig.   56.  —  Cartoon  Effect  to  Show  Lack  of  Playground  Space. 
(From  Newburgh  Survey,  page  63,  by  permission  Of  the  Russell  Sage  Foundation.) 


b.  Ayres's  A  Comparative  Study  of  the  Public  School  Systems  in  the 
Forty-eight  States,  page  6. 

The  value  of  school  property  in  different  states  is  represented  by  in- 
dividual dollar  marks,  a  line  for  each  state,  giving  a  bar  effect ;   thus  : 


Florida 

$15 

Kentucky 

15 

Arkansas 

13 

Mississippi 

4 

Graphic  Presentations  of  School  Statistics    261 


c.   The  1912  Newton,  Massachusetts,  Report,  page  86. 

The  enrollment  of  girls  is  represented  proportionately  for  the  dif- 
ferent kinds  of  schools.  The  lower  grades  are  shown  with  smaller 
children  and  the  high  school  with  large  ones.     But  the  true  relation- 

15  THE  HIGH  TAX  CRY  JUSTIFIED  ? 

Total  Tax  per-  Capita  ,1911 
Mi. Vern  on 

New  Rochelle 

Niagara  Falls 

Average  Jtfcities 

Kingston 

Poughkeepsie 

Jamestown 

Newb  urgh 

Water  town 

Auburn 

Amsterdam 

Fig.  57.  —  Illustration  of  Use  of  a  Bar  Graph  for  Publicity  Purposes. 

Note  that  the  figures  come  at  the  right  ends  of  the  bars,  but  are  much  lighter  in 
effect  and  so  do  not  make  .the  bars  appear  longer.  (From  the  Newburyh  Survey,  page 
97,  by  permission  of  the  Russell  Sage  Foundation.) 

ships  are  shown  by  the  numbers  of  figures  in  the  various  groups. 
This  gets  all  the  value  of  the  cartoon  effect  and  avoids  any  trouble 
about  comparisons  through  areas. 

Time  Charts.  A  special  device  for  comparing  the  use 
of  something  for  several  purposes,  or  the  time  put  in  by- 
different  persons,  is  the  time  chart.     This  is  only  one 


12.48 


262 


School  Statistics  and  Publicity 


form  of  a  distribution  table.  The  time  element  appears 
on  one  scale  and  the  items  on  the  other.  The  data  to 
be  entered  are  put  in  the  rectangles  determined  by  the 
two  scales.  These  areas  may  be  shaded,  or  they  may 
simply  be  filled  with  words.  Such  a  chart  may  be  very 
profitably  used  to  show  how  teachers  spend  their  time, 
and  it  thus  becomes  simply  the  regulation  school  program  ; 
or  it  may  be  used  to  show  how  much  of  the  time  various 
rooms  are  occupied ;  or  it  may  be  utilized  to  show  how 
much  of  the  time  the  school  plant  or  grounds  are  used 
during  a  year,  etc.  Table  42  is  an  example  of  this ; 
the  rooms  could  be  listed  down  the  side  and  the  hours 
at  the  top  if  preferred. 

Table  42.     Time  Chart  by  Rooms  and  Classes 


Hours 

Room  I 

Room  II 

Room  III 

Room  IV 

Room  V 

8 

Math.  1 
Math.  1 

Eng.  4 

Hist.  2 

Sci.  ? 

For.  Lang.  1 

9 

Eng.  1 

10 

Math.  2 

Eng.  2 

Hist.  4 

Sci.  1 

For.  Lang.  2 

11 

1 

Eng.  1 
Eng.  3 

Hist.  3 

"Sci.  2 

2 

Math.  3 

3 

Math.  4 

Hist.  1 

For.  Lang.  3 

Comparisons  with  "  Curves."  It  is  probably  unwise 
to  use  curves  often  in  presenting  statistical  material  to 
the  public.  It  is  well  for  the  superintendent  to  know  how 
to  employ  curves  of  various  kinds,  for  they  are  helpful 
in  his  own  analysis  of  his  data.  Such  uses  have  been 
shown  on  pages  166ff.  Engineers,  draftsmen,  economists, 
social  workers,    and    other    statisticians,  of  course,  use 


Graphic  Presentations  of  School  Statistics    263 

them.  They  are  to  be  seen  in  a  few  advertisements. 
But  they  probably  have  little  significance  for  the  general 
public  as  yet. 

At  present,  curves  can  probably  be  used  most  effectively 
for  the  public  when  the  aim  is  to  make  comparisons  be- 
tween the  changes  in  the 

1909-10.  1910-11.     1911-  12 
$35 


TECH.  HIGH 


NEWTON    HIGH 


same  classes  of  items  on 
different  dates.  For  this 
the  same  vertical  scale 
may  be  used  on  either  side 
of  the  diagram,  with  the 
years  going  out  from  left 
to  right. 

A  good  example  is  found  in 
the  chart  showing  the  decreas- 
ing cost  per  pupil  for  various 
groups,  given  in  Figure  58. 
The  chart  is  very  fine,  except 
that  the  omission  of  the  zero 
line  exaggerates  the  decrease 
altogether  too  much. 

A  still  better  example  of  the 
use  of  curves  for  this  purpose  is 
Figure  59. 

Another  example  is  a  form 
showing  the  amount  of  money 
schools  get  as  compared  with 
what  the  city  as  a  whole 
spends. 

Thus' Professor  Cubberley  gives  two  graphs  to  show  such  facts  for 
two  cities.     (See  Figure  60.) 

Curve  Effects  with  Bar  Graphs.  At  present,  instead  of 
presenting  curves  to  the  public,  it  is  probably  better  to 
use  bar  graphs  arranged  to  secure  the  same  general  effect. 


90 
85 
80 
15 
70 
65 
60 
55 
50 
45 
40 
35 
30 


ALL    PUPILS 


GRADES 


KINDERGARTEN 


Fig.  58.  —  Use  of  Curves  to  Compare 
Changes  in  the  Same  Classes  of  Items 
on  Different  Dates. 

The  figure  shows  decreasing  cost,  per  pupil 
in  various  schools  for  the  years  1909-12. 
Note  omission  of  zero  line.  (From  1912 
Newton,  Mass.,  School  Report,  page  35.) 


$65 
60 
55 
50 
45 
40 

35 

30 
25 

eo 

15 
10 

5 


1907   '08      *09     '10      Ml       '12      '13      '14      '15 

Fig.   59.  —  Use   of   Curves    to  Compare    Changes   in  the  Same  Classes  of 
Items  on  Different  Dates. 

It  shows  cost  per  pupil  in  grades  :md  high  school,  1907-15,  Rockford,  Illinois.     Note 
zero  line  is  given  here.      (From  1915  School  Report,  Rockford,  Illinois,  page  122.) 

264 


/ 

Cos 

\  Pel 

'  Puf 

)il  H 

igh  5 

ichoc 

[} 

/ 

Cos 

it  Per 

Pup 

il  Ele 

ment 

iry  C 

Irade! 

'■)  / 

t' 

Cost 

Per  P 

upil  f 

or  Ge 

neral 

Expe 

\se.[Gt 

ades^ 

US) 

Graphic  Presentations  of  School  Statistics    265 

Where  two  items  are  represented  upon  each  of  a  number 
of  bars,  and  the  part  representing  one  of  these  items  is 
colored  or  shaded,  the  effect  of  two  curves  is  obtained. 
The  graph  as  it  stands  satisfies  the  untrained  reader, 
while  the  trained  reader  can  easily  read  off  the  curves. 
The  diagram  showing  the  retarded  children  in  the  grades 


Total  City  Tax  Rate 


Total  City  Tex  Rate- 


.Schenectady,  NX 


San   Francisco.Cal. 


Fig.  60. 


Example    of    Using    Curves    to    Show   Changes    on    Different 
Dates  in  Two  Items  for  Two  Cities. 


The  two  charts  show  the  competition  for  city  funds.      (From  Cubberley's  Public 
School  Administration,  page  414,  by  permission  of  Houghton  Mifflin  Company.) 


of   Memphis,   Tennessee,   public   schools   affords   a   fine 
illustration.     (See  Figure  22,  page   166.) 

This  same  curve  effect  can  be  obtained  by  a  trained  man 
from  the  component  part  bar  graphs.  Obviously,  if  only 
two  component  parts  are  shown,  the  curve  is  read  one 
way  for  one  set,  and  another  for  the  other,  that  is,  looked 
at  from  the  two  sides  separately,  as  in  Figure  61.     If 


266 


School  Statistics  and  Publicity 


there  are  more  than  two  component  parts,  the  curves 
show  only  for  the  two  end  parts,  as  in  Figure  62. 


Fig.  61.  —  Curve  Effect  on  Bars  with  Two  Component  Parts. 

The  curve  effect  is  also  noticeable  in  a  graph  where 
various  items  in  one  group  are  compared  with  similar 
items  in  other  groups. 

A  good  example  is  the  standing  in  the  four  fundamental  operations 
in  the  Courtis  arithmetic  tests  for  different  grades.  Letting  A  stand 
for  Addition,  S  for  Subtraction,  M  for  Multiplication,  and  D  for 


Fig.   02.  —  Curve  Effect  on  Bars  with  Three  Component  Parts. 

Division,  the  results  might  be  shown  as  in  Figure  63.  A  line  connect- 
ing the  ends  of  all  the  bars  of  the  same  kind  will  give  a  curve.  See 
also  Figure  38. 

Standards  for  Drawing  Curves.  If,  after  all  the  pre- 
ceding, it  is  still  felt  desirable  to  present  school  statistics 
to  the  public  with  curves,  the  curves  should  be  drawn 
properly.     Accordingly,    the    suggestions    of    the    Joint 


Graphic  Presentations  of  School  Statistics    267 


Committee    on    Standards    for    Graphic    Presentation l 
should  be  followed.     The  words  alone  are  given  here,  but 


No. 

16 


0 


probs. 

GRADE  W 


~ 


eic. 


A      5     M     D 


A      S     M      D 


A     S     M     0 


Fig.  63.  —  Bar  Graph   Device  with  Curve  Effect  for  Comparing  Several 
Groups  on  Several  Items. 

Each  bar  represents  the  standing  in  the  Courtis  Tests.  A  means  addition,  S, 
subtraction,  etc. 

the   full   report   contains   graphs   which   make   the   text 
much  clearer : 

1.  The  general  arrangement  of  a  diagram  should  proceed  from 
left  to  right. 

2.  Where  possible,  represent  quantities  by  linear  magnitudes,  as 
areas  or  volumes  are  more  likely  to  be  misinterpreted. 

3.  For  a  curve,  the  vertical  scale,  whenever  practicable,  should  be 
so  selected  that  the  zero  line  will  appear  on  the  diagram. 

4.  If  the  zero  line  of  the  vertical  scale  will  not  normally  appear  on 
the  curve  diagram,  the  zero  line  should  be  shown  by  the  use  of  a  hori- 
zontal break  in  the  diagram. 

5.  The  zero  lines  of  the  scales  for  a  curve  should  be  sharply  dis- 
tinguished from  the  other  coordinate  lines. 

1  Copies  may  be  obtained  from  the  American  Society  of  Mechani- 
cal Engineers,  29  West  39th  Street,  New  York,  price  10  cents,  discount 
in  quantities. 


268  School  Statistics  and  Publicity 

6.  For  curves  having  a  scale  representing  percentages,  it  is  usually 
desirable  to  emphasize  in  some  distinctive  way  the  100  per  cent  line 
or  other  line  used  as  a  basis  of  comparison. 

7.  When  the  scale  of  a  diagram  refers  to  dates,  and  the  period  rep- 
resented is  not  a  complete  unit,  it  is  better  not  to  emphasize  the  first 
and  last  ordinates,  since  such  a  diagram  does  not  represent  the  begin- 
ning or  end  of  time. 

8.  When  curves  are  drawn  on  logarithmic  coordinates,  the  limiting 
lines  of  the  diagram  should  each  be  at  some  power  of  ten  on  the  log- 
arithmic scales. 

9.  It  is  advisable  not  to  show  any  more  coordinate  lines  than 
necessary  to  guide  the  eye  in  reading  the  diagram. 

10.  The  curve  lines  of  a  diagram  should  be  sharply  distinguished 
from  the  ruling. 

11.  In  curves  representing  a  series  of  observations,  it  is  advisable, 
whenever  possible,  to  indicate  clearly  on  the  diagram  all  the  points 
representing  the  separate  observations. 

12.  The  horizontal  scale  for  curves  should  usually  read  from  left 
to  right  and  the  vertical  scale  from  bottom  to  top. 

13.  Figures  for  the  scales  of  a  diagram  should  be  placed  at  the  left 
and  at  the  bottom  or  along  the  respective  axes. 

14.  It  is  often  desirable  to  include  in  the  diagram  the  numerical 
data  or  formulae  represented. 

15.  If  numerical  data  are  not  included  in  the  diagram,  it  is  desirable 
to  give  the  data  in  tabular  form  accompanying  the  diagram. 

16.  All  lettering  and  all  figures  on  a  diagram  should  be  placed  so 
as  to  be  easily  read  from  the  base  as  the  bottom,  or  from  the  right- 
hand  edge  of  the  diagram  as  the  bottom. 

17.  The  title  of  a  diagram  should  be  made  as  clear  and  complete 
as  possible.  Sub-titles  or  descriptions  should  be  added  if  necessary 
to  insure  clearness. 

3.    Special  Summarizing  Graphs 

Sometimes  it  is  desired  to  give  a  graphic  summary  of 
something  that  has  been  measured  by  relative  position 
on  a  number  of  items.  The  way  of  showing  this  roughly 
by  the  bar  graph  was  given  on  page  244,  but  this  indicated 
only  the  quarter  of  the  distribution  in  which  the  case  fell 


Graphic  Presentations  of  School  Statistics    269 


Expenditure 

per 

inhabitant 


.  1 

1 

wsnm 

Z 

3 

3 

4- 

4- 

5 

5 

6 

6 

7 

7 

9 

8 

3 

9 

IO 

1  O 

1  1 

1  1 

/  2. 

m  2  ■ 

13 

1  3 

14 

;  4- 

15 

15 

16 

16 

1  7 

1  7 

1  8 

1  8 

19 

13 

20 

20 

21 

21 

Expenditure  per 
$1,000  of  tax- 
able wealth 


IO 


i  i 


12 


15 


I  6 


17 


19 


20 


21 


Expenditure  per 

child  in  average 

daily  attendance 

Thp   shaded    rectangles    represent  Boston. 


Fig.   04.  —  Graphic  Device   for   Summarizing   the   Relative  Position   of   a 
Given  Case  in  a  Number  of  Different  Distributions. 

This  figure  shows  the  rank  of  Boston  in  a  g'oup  of  twenty-one  cities  in  expenditure 
for  operation  and  maintenance  of  schools.      (From  Boston  Report,  191G,  page  158.) 


270 


School  Statistics  and  Publicity 


Teacher City 

(Indicate  sex) 


EFFICIENCY    RECORD 

Grade  taught. 


(or  building)  (or  subject) 

Experience years.     Salary per  month. 

Highest  academic  training 

Extent  of  professional  training 


Detailed  Rating V.P.   Poor 


T. 


II. 


1.  General  appearance 

2.  Health 

3.  Voice 

4.  Intellectual  capacity 

5.  Initiative  and  self-reliance 

6.  Adaptability  and  resourcefulness  . 

7.  Accuracy 

8.  Industry 

9.  Enthusiasm  and  optimism    . 

10.  Integrity  and  sincerity 

11.  Self-control 

12.  Promptness 

13.  Tact 

.  14.  Sense  of  justice 

'  15.  Academic  preparation 

16.  Professional  preparation 

17.  Grasp  of  subject-matter 

18.  Understanding  of  children    .      .      .      . 

19.  Interest  in  the  life  of  the  school     . 

20.  Interest  in  the  life  of  the  community 

21.  Ability  to  meet  and  interest  patrons 

22.  Interest  in  lives  of  pupils      .      .      .      . 

23.  Co-operation  and  loyalty      .      .      .      . 

24.  Professional  interest  and  growth    . 

25.  Daily  preparation 

.  20.  Use  of  English  

127.  Care  of  light,  heat,  and  ventilation    . 

28.  Neatness  of  room 

29.  Care  of  routine 

30.  Discipline  (governing  skill)  .      .      .      . 

31.  Definiteness  and  clearness  of  aim  . 

32.  Skill  in  habit  formation 

33.  Skill  in  stimulating  thought       .      .      . 

34.  Skill  in  teaching  how  to  study  .      .      . 

35.  Skill  in  questioning 

30.  Choice  of  subject-matter       .      .      .      . 

37.  Organization  of  subject-matter 

38.  Skill  and  care  in  assignment 

39.  Skill  in  motivating  work        .      .      .      . 
{  40.  Attention  to  individual  needs    . 
r 41.  Attention  and  response  of  the  class    . 
j  42.  Growth  of  pupils  in  subject-matter     . 

43.  General  development  of  pupils 

!  44.  Stimulation  of  communitv     .      .      .      . 

i  45.  Moral  influence  ........ 


IV. 


General  Patino 


Medium 


Good   Ex, 


Recorded  by Position Date 

Fig.  65.  —  Stinmmrizine:  Graph   to  Show  Efficiency  Record  of  a  Teacher, 
Used  by  School  of  Education,  University  of  Chicago. 


Graphic  Presentations  of  School  Statistics    271 

on  that  quality.     A  refinement  of  this  is  found  in  Figure 
64. 

Another  example  is  that  used  by  the  School  of  Education 
at  the  University  of  Chicago  to  sum  up  the  rating  of  a 


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55 

59 

G+ 

70    SO 

25 


Grade       n         m       ez        2        w       m      2m' 

Arithmetic 

Addition 

Speed 

Accuracy 

Subtraction 

Speed 

Accuracy 

Multiplication 

Speed 

Accuracy 

Division 

Speed 

Accuracy 

Silent  Reading 
Handwriting 
Speed 
Quality 


Grade      n         in        nr        n        3a       m       mn 

Fig.  Of).  —  Graphic   Device    for    Summarizing    the    Achievements    of    One 
Pupil  or  School  in  Several  Fields  as  Related  to-Standards  in  Those  Fields. 

(From  Educational  Tents  ant!  Measurements  of  Monroe,  De  Voss  and  Kelly,  by 
permission  of  the  authors  and  Houghton  Mifllin  Company.) 

teacher  on  several  different  qualities.    Part  of  this  graph  is 
shown  in  Figure  65. l 

It  should  be  noted  that  a  general  heading  will  loom 
up  on  this  graph  in  proportion  to  the  number  of  sub- 
heads it  has.  Consequently,  the  way  to  make  any  general 
heading  have  weight  will  be  to  add  sub-heads,  without 
caring  particularly  how  important  these  sub-heads  are. 

1  The  original  blank  has  suitable  main  headings  for  each  Roman  numeral, 
printed  horizontally,  but  it  was  not  feasible  to  show  them  on  a  page  of  this 
size. 


272  School  Statistics  and  Publicity 

There  appears  to  be  no  way  of  overcoming  this  defect 
except  by  varying  the  widths  of  the  horizontal  spaces, 
which  would  probably  complicate  the  device  beyond  the 
point  of  practical  value. 

A  third  summarizing  device,  and  one  capable  of  wide 
adaptation,  is  shown  in  Figure  66.  The  same  device 
may  be  used  to  show  the  standing  of  a  school  or  city  on 
a  number  of  items,  by  placing  the  names  of  the  standards 
at  the  top  in  place  of  the  Roman  numerals  for  the  grades. 

Figure  24  on  page  168,  showing  the  surfaces  of  fre- 
quency one  above  the  other,  is  a  fourth  summarizing 
graph  that  is  of  value  to  the  public. 

4.    Brinton' s  Rules  for  Graphic  Presentation 

Brinton,  in  his  standard  book,  Graphic  Methods  of 
Presenting  Facts,  has  a  very  convenient  set  of  rules 
followed  by  a  set  of  check  items.  These  were  printed 
before  the  suggestions  of  the  Committee  on  Standards 
for  Graphic  Presentation  (of  which  he  is  chairman)  were 
published  (see  pages  267-268),  and  some  of  the  suggestions 
appear  in  both  sets.  But  Mr.  Brinton's  original  lists 
are  not  confined  to  suggestions  for  curves,  as  is  the  report 
of  the  committee.  The  practical  school  man  will  prefer 
to  have  all  the  suggestions  given  in  one  place,  so  they  are 
here  appended. 

Helps  on  Graphic  Presentations 

(Selected  from  Brinton  :   Graphic  Methods  of  Presenting  Facts, 
pages  3fi0  362) 
I.    Ralls  for  Graphic  Presentation 

1.  Avoid  using  areas,  or  volumes,  when  representing  quanti- 
ties. Presentations  read  from  only  one  dimension  are  the  least 
likely  to  be  misinterpreted. 


Graphic  Presentations  of  School  Statistics    273 

2.  The  general  arrangement  of  a  chart  should  proceed  from  left 
to  right. 

3.  Figures  for  the  horizontal  scale  should  always  be  placed  at 
the  bottom  of  a  chart.  If  needed,  a  scale  may  be  placed  at  the 
top  also. 

4.  Figures  for  the  vertical  scale  should  always  be  placed  at 
the  left  of  a  chart.  If  needed,  a  scale  may  be  placed  at  the  right 
also. 

5.  Whenever  possible,  include  in  the  chart  the  numerical  data 
from  which  the  chart  was  made. 

6.  If  numerical  data  cannot  be  included  in  the  chart,  it  is  well 
to  show  the  numerical  data  in  tabular  form  accompanying  the  chart. 

7.  All  lettering  and  all  figures  on  a  chart  should  be  placed  so 
as  to  be  read  from  the  base  or  from  the  right-hand  edge  of  the 
chart. 

8.  A  column  of  figures  relating  to  dates  should  be  arranged  with 
the  earliest  date  at  the  top. 

9.  Separate  columns  of  figures,  with  each  column  relating  to  a 
different  date,  should  be  arranged  to  show  the  column  for  the 
earliest  date  at  the  left. 

10.  When  charts  are  colored,  the  color  green  should  be  used  to 
indicate  features  which  are  desirable  or  which  are  commended, 
and  red  for  features  which  are  undesirable  or  criticized  adversely. 

11.  For  most  charts  and  for  all  curves,  the  independent  variable 
should  be  shown  in  the  horizontal  direction. 

12.  As  a  general  rule,  the  horizontal  scale  for  curves  should 
read  from  left  to  right  and  the  vertical  scale  from  bottom  to  top. 
(See  "special.") 

13.  For  curves  drawn  on  arithmetically  ruled  paper,  the  ver- 
tical scale  whenever  possible  should  be  so  selected  that  the  zero 
line  will  be  shown  on  the  chart. 

14.  The  zero  line  of  the  vertical  scale  for  a  curve  should  be  a 
much  broader  line  than  the  average  coordinate  lines. 

15.  If  the  zero  line  of  the  vertical  scale  cannot  be  shown  at  the 
bottom  of  a  curve  chart,  the  bottom  line  should  be  a  slightly 
wavy  line  indicating  that  the  field  has  been  broken  off  and  does 
not  reach  to  zero. 

16.  When  the  scale  of  a  curve  chart  refers  to  percentages,  the 
line  at  100  per  cent  should  be  a  broad  line  of  the  same  width  as  a 
zero  line. 


274  School  Statistics  and  Publicity 

18.  If  the  horizontal  scale  for  a  curve  begins  at  zero,  the  verti- 
cal line  at  zero  (usually  the  left-hand  edge  of  the  field]  should  be 
a  broad  line. 

19.  When  the  horizontal  scale  expresses  time,  the  lines  at  the 
left-hand  and  the  right-hand  edges  of  a  curve  chart  should  not 
be  made  heavy,  since  a  chart  cannot  be  made  to  include  the  be- 
ginning or  the  end  of  time. 

20.  When  curves  are  to  be  printed,  do  not  show  any  more 
coordinate  lines  than  are  necessary  for  the  data  and  to  guide  the 
eye.     Lines  one-fourth  inch  apart  are  sufficient  to  guide  the  eye. 

21.  Make  curves  with  much  broader  lines  than  the  coordinate 
ruling,  so  that  the  curves  may  be  clearly  distinguished  from  the 
background. 

22.  Whenever  possible,  have  a  vertical  line  of  the  coordinate 
ruling  for  each  point  plotted  on  a  curve,  so  that  the  vertical  lines 
may  show  the  frequency  of  the  data  observations. 

23.  If  there  are  not  too  many  curves  drawn  in  one  field,  it  is 
desirable  to  show  at  the  top  of  the  chart  the  figures  representing 
the  value  of  each  point  plotted  in  a  curve. 

24.  When  figures  are  given  at  the  top  of  a  chart  for  each  point 
in  a  curve,  have  the  figures  added  if  possible  to  show  yearly  totals 
or  other  totals  which  may  be  useful  in  reading. 

25.  Make  the  title  of  a  chart  so  complete  and  so  clear  that 
misinterpretation  will  be  impossible. 

Special. 

In  showing  deviations  from  a  central  tendency,  on  the  vertical 
scale,  upwards  is  plus,  and  downwards,  minus;  on  the  horizontal 
scale,  to  the  right  is  plus,  and  to  the  left,  minus. 

II.    Checking  List  for  Graphic  Presentations. 

1.  Are  the  data  of  a  chart  correct? 

2.  Has  the  best  method  been  used  for  showing  the  data? 

'■I.  Are  the  proportions  of  the  chart  the  best  possible  to  show 
the  data? 

4.  When  the  chart  is  reduced  in  size,  will  the  proportions  be 
those  best  suited  to  the  space  in  which  it  must  be  printed? 

5.  Are  the  proportions  such  that  there  will  be  sufficient  space  for 
the  title  of  the  chart  when  the  chart  has  been  reduced  to  final 
printing  size? 


Graphic  Presentations  of  School  Statistics    275 

G.    Are  all  scales  in  place? 

7.  Have  the  scales  been  selected  and  placed  in  the  best  pos- 
sible manner? 

8.  Are  the  points  accurately  plotted? 

9.  Are  the  numerical  figures  for  the  data  shown  as  a  portion  of 
the  chart? 

10.  Have  the  figures  for  the  data  been  copied  correctly? 

11.  Can  the  figures  for  the  data  be  added  and  the  total  shown? 

12.  Are  all  the  dates  accurately  shown? 

13.  Is  the  zero  of  the  vertical  scale  shown  on  the  chart? 

14.  Are  all  zero  lines  and  the  100  per  cent  lines  made  broad 
enough  ? 

15.  Are  all  lines  on  the  chart  broad  enough  to  stand  the  re- 
duction to  the  size  used  in  printing? 

16.  Does  lettering  appear  large  enough  and  black  enough  when 
seen  under  a  reducing  glass  in  the  size  which  will  be  used  for  print- 
ing? 

17.  Is  all  the  lettering  placed  on  the  chart  in  the  proper  direc- 
tion for  reading? 

18.  Is  cross-hatching  well  made  with  lines  evenly  spaced? 

22.  Are  dimension  lines  used  wherever  advantageous? 

23.  Is  a  key  or  legend  necessary? 

24.  Does  the  key  or  legend  correspond  with  the  drawing? 

25.  Is  there  a  complete  title,  clear  and  concise? 

26.  Is  the  drafting  work  of  good  quality? 

27.  Have  all  pencil  lines  which  might  show  in  the  engraving 
been  erased? 

28.  Is  there  any  portion  of  the  illustration  which  should  be 
cropped  off  to  save  space? 

29.  Are  the  instructions  for  the  final  size  of  the  plate  so  given 
that  the  engraver  cannot  make  a  mistake? 

3^0.    Is  the  chart  in  every  way  ready  to  mark  "O.  K. "? 

5.    Presenting  Statistical  Data  with  Maps 

Sometimes  it  is  necessary  to  impress  the  public  with  the 
way  items  vary  in  size  or  frequency  in  different  geographic 
areas.     Thus,  a  state  superintendent  may  wish  to  show 


276  School  Statistics  and  Publicity 

the  legislature  how  high  school  facilities  vary  in  different 
counties  of  the  state ;  a  city  superintendent  may  desire 
to  demonstrate  how  far  pupils  have  to  go  to  reach  a 
school,  or  the  crowded  conditions  necessitating  a  new 
building  in  a  certain  locality ;  or  a  county  superintendent 
may  need  to  show  just  how  his  county  ought  to  be  dis- 
tricted for  schools  so  that  all  children  will  be  within  a 
reasonable  distance  of  a  school.  These  things  may  be 
shown  fairly  well  by  variations  in  shading  the  maps  for 
different  localities,  a  device  much  used  by  the  United 
States  Bureau  of  Census  and  by  geography,  history,  and 
economics  texts.  However,  all  such  maps  require  keys  for 
their  interpretation,  and  it  is  difficult  for  any  one,  except 
a  person  very  familiar  with  such  work,  to  understand 
them  quickly. 

A  much  better  way  is  to  represent  every  case  or  every 
certain  number  of  cases  (say  ten)  by  a  dot.  Figure  67 
is  a  map  used  by  George  Peabody  College  for  Teachers 
to  show  the  communities  from  which  students  have 
come  to  the  college  during  the  years  1914  to  1918.  This 
illustration  shows  admirably  just  how  much  territory  is 
coming  under  the  direct  influence  of  the  school.  It  does 
not  give  a  correct  notion  of  the  number  of  students 
that  have  attended  the  college  during  these  years,  as 
some  cities  have  contributed  possibly  a  hundred  or  more 
each  and  yet  such  a  city  would  be  represented  by  only 
one  dot.  If  a  dot  had  been  inserted  for  each  student, 
some  portions  of  the  map  would  have  been  solid  black. 
This  could  be  remedied  by  using  perpendicular  wires 
with  a  bead  for  each  student,  as  recommended  by  Brinton,1 
but  it  is  very  difficult  to  reproduce  such  a  map  by  pho- 
tography or  by  a  drawing.     In  showing  the  widespread 

1  -draplilc  Metfiods  of  Preseutiny  Facts,  p.  251 


277 


278 


School  Statistics  and  Publicity 


influence  of  the  college,  the  map  is  more  effective  than 
the  one  from  the  Report  of  the  General  Education  Board 


Fig.  08.  —  Device  for  Showing  Distribution  of  Cases  on  a  Map. 
Each  dot  represents  a  student  from  the  different  counties,  attending  the  University 
of  Georgia.      Note   the   radiating   lines   to  indicate  that  the  university  is   exerting   ar 
influence  upon  all  the  state. 

for  1914,  page  4.     This  is  merely  a  map  of  the  United 
States  in  outline,  with  the  number  of  students  in  each 


Graphic  Presentations  of  School  Statistics    279 

state  attending  Vanderbilt  University.  It  is  thus  only  a 
map  with  a  table  on  it,  and  not  even  in  good  tabular 
form,  because  there  is  no  direct  way  to  compare  the 
numbers  by  having  them  in  one  column,  and  especially 
running  from  low  to  high  or  vice  versa.  It  is  impossible 
for  one  to  visualize  the  number  of  students  attending 
Vanderbilt  University,  and  it  takes  a  little  time  to  get  a 
correct  notion  of  the  sections  of  the  country  coming  under 
its  direct  influence. 

The  University  of  Georgia  uses  a  good  map  to  show  its 
attendance,  heightening  the  effect  by  adding  radiating 
lines  to  indicate  its  influence  in  the  state.     (See  Figure  68.) 

The  dot  device  on  a  map  of  the  city  was  used  by  the 
superintendent  in  the  Rockford  schools  l  to  show  the 
residence  of  pre-tubercular  pupils  and  also  the  residence 
of  students  in  the  evening  schools.  In  the  one  case,  he 
showed  clearly  that  the  pre-tubercular  children  were 
scattered  widely  through  the  city  and  that  the  problem 
of  handling  them  was  city-wide.  He  also  showed  that 
the  attendance  at  evening  school  was  not  restricted  to  a 
small  area  of  the  city.  In  some  of  the  Red  Cross  work 
huge  state  maps  are  shown  with  the  number  of  tubercular 
soldiers  for  each  county  pasted  in  as  so  many  paper  doll  sol- 
diers, upright  in  ranks.     It  is  a  very  effective  presentation. 

If  the  distances  on  a  map  are  to  be  measured  in  some 
time  unit,  a  map  similar  to  that  in  Figure  69  is  useful. 

The  accredited  schools  of  a  state  may  be  shown  on  an 
outline  map,  using  different-colored  yjins  or  tacks  for  the 
different  classes  of  schools.  In  the  division  of  rural 
education  in  the  state  department  of  education  for 
Missouri,  there  is  an  immense  map  of  the  state  painted 
on  the  wall.  On  this,  each  approved  rural  school  is  shown 
1  Rockford  Review  for  191 5-16,  pp.  85,  104 


280 


School  Statistics  and  Publicity 


by  a  small  kodak  picture  of  the  building.  This  enables  a 
visitor  to  get  almost  immediately  an  idea  of  where  the  good 
rural  school  work  is  being  done,  and  closer  examination 
will  show  the  kinds  of  school  buildings  being  put  up.     Of 


Fro.  GO.  — Device  for 


wing  Distance  with  ;i  Time  Element  on  a  Map. 


Map   used    l>\   the  Nashville  Commercial  Club  to  show  how  accessible   the  city  is. 
Every  city  indicated  is  within  twelve  hours'  travel  of  Nashville. 

course,  such  a  map  cannot  be  easily  reproduced  in  printing. 
Still,  something  can  be  done  with  conventional  drawings 
for  each  item  in  a  class.  Most  readers  of  this,  for  example, 
probably  recall  the  recent  Y.  M.  C.  A.  propaganda  with 


Graphic  Presentations  of  School  Statistics    281 

the  sketch  of  the  Western  Battle  Front,  each  building  of 
the  organization  being  represented  by  a  tiny  drawing  of 
the  right  kind. 

III.   HOW  GRAPHS  FOR  THE  PUBLIC  DIFFER  FROM  THOSE 
FOR  THE   ADMINISTRATOR 

The  difference  has  been  referred  to  many  times  before, 
especially  in  discussion  of  the  superior  value  for  the 
public  of  bar  graphs  over  curves  and  in  Brinton's  check 
list.1  Let  us  now  analyze  it  further,  restating  certain 
points  for  additional  emphasis. 

In  general,  the  public  will  view  graphs  in  much  the 
same  way  as  they  view  any  explanation  or  presentation. 
The  ordinary  man  cannot  quickly  get  from  a  rough  copy 
of  a  chapter  the  meaning  that  an  experienced  writer  can ; 
he  cannot  extract  from  a  confused  and  verbose  mass  of 
evidence  the  essentials  that  a  trained  lawyer  can ;  he 
cannot  grasp  so  quickly,  nor  in  such  large  numbers, 
many  intricate  graphic  presentations  that  seem  relatively 
simple  to  a  trained  school  man.  School  graphs  for  the 
public  must  be  simple,  with  relatively  few  elements  or 
lines,  and  very  forcible.  The  trained  reader  can  extract 
the  significant  things  from  most  graphs,  however  poorly 
constructed  ;  the  average  man  cannot.  Let  us  now  take 
up  some  of  the  most  significant  aids  for  making  graphs 
clear  to  the  public. 

1.  The  title  should  give  all  the  significant  points  to  be 
found  ih  the  graph,  so  that  the  graph  would  be  quite  in- 
telligible apart  from  the  context  where  it  is  found. 

This  implies  that  there  should  not  be  many  different  points  found 
in  one  chart.  If  the  chart  can  show  only  one  thing,  it  is  all  the  better. 
Some  examples  of  good  titles  are : 

1  See  pp.  274-5 


282  School  Statistics  and  Publicity- 

standing  of  the  children  of  Salt  Lake  City  in  the  fundamentals 
of  arithmetic,  judged  by  the  median  score  attained  by  each  grade. 
(Salt  Lake  City  Survey,  p.  175.) 

Distribution  of  ages  at  which  Salt  Lake  City  children  enter  the 
first  school  grade.      (Same,  p.  201.) 

C  olumns  represent  number  of  pupils  among  each  hundred  begin- 
ners who  remain  in  school  at  each  grade  from  the  first  elementary 
to  the  fourth  high.     (Cleveland  Survey,  p.  88.) 

Percentage  of  elementary  teachers,  high  school  teachers,  and 
elementary  principals  in  Cleveland  who  are  home  trained  and  not 
home  trained.     (Cleveland  Survey,  p.  107.) 

How  Portland  spends  its  dollar.  (Portland  Survey,  p.  84.) 
Figure  7,  representing  the  percentage  of  children  in  several 
grades  who  make  the  given  scores  in  composition.  For  ex- 
ample, 1.7  per  cent  of  the  fourth-grade  children  wrote  com- 
positions scored  at  0;  43.8  per  cent  of  the  fourth  grade  were 
scored  at  1 ;  etc.  By  following  the  median  lines,  the  overlap- 
ping of  ability  from  grade  to  grade  is  disclosed.  (Butte  Survey, 
p.  75.) 

In  each  of  these  instances,  the  title  and  the  chart  make  a  complete 
unit.  The  last  one  is  especially  noteworthy.  It  gives  a  very  brief 
title,  then  expands  this  with  a  full  but  concise  explanation  of  all  points 
in  the  diagram  that  may  cause  confusion. 

2.  A  chart  or  graph  too  large  to  be  seen  without  turning 
the  head  is  apt  to  be  a  poor  chart  for  the  public,  no  matter 
how  simple  it  may  be. 

This  is  shown  by  some  charts  and  folded  graphs  in  some  of  the 
surveys.  It  is  very  seldom  that  a  chart  or  graph  should  occupy  more 
than  one  page  of  the  publication  in  which  it  appears. 

In  reducing  charts  for  publication,  however,  one  must  be  careful 
that  the.  reduction  is  not  carried  to  the  point  of  making  the  differences 
negligible  or  the  lettering  too  small  to  be  read  easily.  Advertisers 
1  )  lg  ago  discovered  that  the  public  will  not  read  advertisements  in 
fine  print,  and  school  graphs  are  only  a  form  of  advertising  the  school 
work.  Often  graphs  are  reduced  greatly  to  economize  space.  If 
this  is  pushed  to  the  extent  of  making  the  graph  hard  to  read,  it  is 
clearly  advisable  to  omit  some  of  the  graphs  altogether,  and  make  the 
remainder  large  enough  to  be  effective. 


Graphic  Presentations  of  School  Statistics    283 

3.  The  background  of  a  chart  should  not  be  made  any 
more  prominent  than  necessary. 

Many  charts  are  plotted  on  coordinate  paper  heavily  and  finely 
ruled,  while  the  curves  or  bars  are  but  a  trifle  heavier  than  the  co- 
ordinate ruling.  Such  charts  do  not  stand  out  clearly  from  their 
background.  Only  as  many  coordinate  lines  should  appear  on  a 
chart  as  are  necessary  to  guide  the  eye  of  the  reader  and  to  permit  of 
easy  reading  of  the  curves. 

The  difficulty  may  be  rather  easily  avoided  by  drawing  on  the 
coordinate  paper  in  very  heavy  lines  with  India  ink  all  the  lines  and 
figures  which  it  is  desired  to  reproduce,  including  the  coordinate 
ruling.  The  necessary  lettering  can  be  put  in  with  a  typewriter,  using 
a  practically  new  black  ribbon.  The  proper  exposure  in  making  the 
cut  will  "take"  all  the  desired  lines  and  lettering,  but  not  the  others. 

4.  Exaggerations  should  be  avoided  as  much  as  possible. 

(a)  Usually,  a  very  forceful  presentation  may  be  had  without  any 
great  sacrifice  of  accuracy. 

Of  course,  only  complete  scientific  presentation  can  ever  give 
the  whole  truth.  However,  if  only  a  phase  of  a  problem  at  a 
time  is  presented  to  the  public  (and  often  this  seems  necessary), 
some  exaggeration  is  inevitable.  Here  the  problem  is  to  choose 
between  absolute  accuracy  and  forcefulness  of  presentation. 

(b)  It  is,  in  general,  dangerous  to  leave  the  zero  line  off  a  chart 
intended  for  the  public,  or  even  to  send  it  out  with  the  conventional 
wave  line  at  the  bottom. 

Many  school  men  in  making  charts  have  found  it  convenient 
to  leave  the  zero  line  off.  Sometimes,  when  all  the  quantities 
used  come  high  on  the  vertical  scale,  this  is  done  to  economize 
space. 

An  excellent  example  of  such  a  chart  is  Figure  70.  The 
upper  chart  conveys  the  idea  that  salaries  have  increased  greatly 
in  Louisville  during  this  period  of  five  years.  But  the  chart 
begins  at  about  $475  instead  of  zero.  The  chart  below  is  drawn 
in  full.  It  is  clearly  seen  from  the  complete  chart  that  the  salary 
increase,  while  noticeable1,  docs  not  appear  anything  like  so  large 
as  in  the  original  and  incorrectly  drawn  chart. 


284 


School  Statistics  and  Publicity 


(c)  Care  should  be  taken  that  the  scales  chosen  for  the  graph  do 
not  exaggerate  things  unduly. 

The  novice  in  chart  making  will  probably  become  confused  by 
the  ever  changing  ratios  between  the  perpendicular  and  the  hori- 
zontal scales.     No  definite  rules  can  be  laid  down  for  guidance  in 

As  if  wds  drawn 


<550   #600  *650  $700 

I9IO-H   ] 

1911^121 

19)3-14   1 

As 

t  should 

have 

been 

0     J5"0  -f|O0  ^150  4200  4*250  ^300  4350  fA 00 $4 SO  4 500  $550  4600  ^650  #700 

I9I0J-II 

1811 

-\z 

I9I2[-I3 

K 

1913 

-K 

1914 

-IS 

I'm 


<(). 


Example  of  Danger  of  Leaving  Zero  Line  Off  a  Chart. 


The  top  set  of  bars  was  intended  to  show  a  "comparison  of  salaries  paid  to  elementary 
school  teachers  in  Louisville  for  the  years  1910-1915."  The  bottom  set  shows  what  the 
comparison  really  was.      (Adapted  from  the  Louisville  Report,  1914-1915,  page  IS.) 


this  matter.  The  only  way  to  get  facility  in  adjusting  these 
ratios  in  the  proper  way  is  through  the  trial  and  error  method. 
That  a  change  in  the  ratio  makes  a  great  difference  in  the  im- 
pression produced  by  the  graph  is  shown  in  Figure  71 


Graphic  Presentations  of  School  Statistics    285 

The  left-hand  graph  shows  the  results  of  the  Kansas  Silent 
Reading  Tests  in  the  Rockford  schools.  The  right-hand  graph 
shows  the  same  data  plotted  with  the  horizontal  scale  increased 
while  the  vertical  scale  is  decreased.     It  will  be  seen  at  once 


Scale    Grade 

345  6  7  8 
20 


15 


10 


5     ■ 


Scale  Grade 

3       4       5        6       7      8 

HO 


15 


10 


Fig.  71.  —  Example  of  Effect  Produced  by  Changing  Ratio  of  Horizontal  to 
Vertical  Scale  on  a  Graph. 

Results  of  Kansas  Silent  Reading  Tests.      (Adapted  from  Review  of  Rockford,  Illinois, 
Schools,  [«n.j-l<)10.) 


286  School  Statistics  and  Publicity 

that  the  difference  in  achievement  between  the  grades  does  not 
appear  so  marked  in  the  second  illustration  as  in  the  first. 

(d)  The  superintendent  must  beware  of  graphs  containing  optical 
illusions. 

There  is  little  danger  of  this  in  charts  made  small  enough  to 
publish,  but  there  may  be  danger  in  large  wall  charts.  Brinton 
calls  attention  to  illusions  caused  by  a  row  of  perpendicular  lines 
compared  with  a  row  of  horizontal  ones  the  same  distance  apart.1 


Fig.  72.  —  Cartoon  Graph  Representing  Ratio  of  Lighting  Space  to  Floor 

Space. 
(From  Alabama  Three-County  Survey,  page  92.) 

The  lines  in  the  first  row  appear  shorter  than  they  really  are  and 
spread  farther  apart ;  those  in  the  second  seem  to  be  longer  than 
they  really  are.  Another  illusion  is  caused  when  a  white  square 
and  a  black  one  of  exactly  the  same  size  are  drawn  adjacent.  The 
white  one  seems  larger.  This  might  affect  slightly  the  ratio  of 
lighting  space  as  shown  by  white  windows  against  a  dead  black  wall 
space,  in  some  surveys.     (See  Figure  72.) 

5.  Special  effort  should  be  made  to  introduce  variety, 
novelty,  and  various  striking  features  to  attract  attention 
to  the  statistical  relations  to  be  presented. 

Variety  in  graphs  is  as  necessary  to  keep  the  attention  as  is  variety 
anywhere  else.  No  ordinary  reader  could  stand  it  to  wade  through 
a  report  of  any  length  which  had  a  great  many  bar  graphs  of  one 
pattern  and  no  other  illustrations. 

Sometimes  pictures  or  devices  may  be  used  simply  to  catch  the 
attention.  The  bulletin  on  illiteracy  in  Virginia,2  for  example,  uses 
a  picture  of  a  rural  school  to  get  the  reader's  attention  for  the  state- 

'  Graphic  Methods  of  Presenting  Facts,  p.  358 

2  Illiteracy  in  Virginia,  published  by  State  Department  of  Public 
Instruction,  p.  7 


Graphic  Presentations  of  School  Statistics    287 


THE  PROFIT  FROM  TWO  HERDS  FOR  ONE  YEAR 


KA 


SyatxBawk         ": 
QLtov*    :    * 95.73 


m 


State  Bank 


WHY  THIS   DIFFERENCE? 

Herd  A 


IT  WAS  NOT  THE  SIZE   OF  HERD II  COWS 

2  pure  Bred 

It  was  mot  the  breed %  grades 

it  was  mot  the  feed  cost *526.70 

(silos  and  good  buildings  on  each  farm) 

HERE  IS  THE  ANSWER 

AVERAGE  PRODUCTION  OF 

BUTTER  FAT Wl.t  LBS. 

PER  COW 

This  Is  A  True  Story  As  Told  By 


Herd  D 


ii  cows 

I  NATIVE 
10  GRADES 

*569.96 


386.9  LBS 
PER  COW 


Moral:- It  would  have  taken  93  poor  cows  to  make 
the  profit  the  ii  good  cows  made. 

DOES  IT  PAY  TO  KEEP  RECORDS? 


Fig.  78.  —  Exampl 


of  Use  of  Special  Devices  to  Attract  Attention  to  the 
Statistic-  Involved. 

This  figure  compares  the  profit  from  two  herds  for  one  year  and  shows  how  many 
dairymen  are  wasting  time  and  money  on  low-producing  cows.      "Why  not  get  rid  of 

your  'visitors".'"      (From  a  publication  of  the  University  of  Wisconsin,  Experiment 
Division,  by  permission.) 


288 


School  Statistics  and  Publicity 


ment  that  only  six  of  these  children  are  beyond  the  first  reader  and 
none  beyond  the  fifth  reader. 

Or  devices  similar  to  those  in  Figure  73  may  be  used  to  attract 
attention  to  the  statistics. 

Then  there  are  special  touches  which  no  one  can  tell  precisely  how 
to  go  about  acquiring.     For  example,  in  the  Alabama  three-county 


One  out  of  every  ten  white  men  must  ask  another 
to  mark  his  ballot  for  him. 

flMSttit 

Four  out  of  every  ten  negroes  must  ask  another 
to  51911  their  names  forthern. 

Fig.  74.  —  Bar  Graph  with  Cartoon  Effect  Showing  Illiteracy  in  Alabama, 
(From  the  Survey  of  Three  Counties  in  Alabama,  page  19.) 

survey,  illiteracy  is  shown  among  voters  by  having  the  literate  voters 
face  the  reader  and  the  illiterate  ones  turn  their  backs.  (See  Figure  74. ) 
Probably  this  is  about  as  forceful  a  showing  Of  the  shame  of  illiteracy 
as  could  be  devised. 

IV.    EXAMPLES  OF  GOOD  GRAPHS  ON  SCHOOL  STATISTICS 
FOR   THE    PUBLIC 

1.    To  Show  Rise  in  School  Costs 

In  the  Newton,  Massachusetts,  Report  for  1912, J  the 
scale  on  tax  for  school  maintenance  per  $1000  is  shown 
on  a  thermometer  device  which  is  reproduced  on  page  102 

1  Page  113 


Graphic  Presentations  of  School  Statistics    289 

of  this  book.  This  gives  the  idea  that  costs  for  school 
maintenance  should  rise.  The  names  of  various  cities 
with  which  Newton  is  compared  appear  at  one  side  of  the 
graph,  with  lines  running  from  each  name  to  the  proper 
degree  on  the  thermometer  where  the  mercury  should 
stand  for  that  city.  This  graph  shows  very  forcibly  that 
the  mercury  must  rise  many  degrees  for  Newton  before 
it  will  equal  the  best  record  made  by  the  other  cities. 

For  some  cities,  probably  a  cartoon  utilizing  the  weigh- 
ing machine  seen  at  fairs  and  carnivals  would  be  equally 
effective.1  Instead  of  pounds,  tax  levies  or  per  capita 
amounts  of  money  could  be  shown  on  the  scale  of  the 
upright,  with  the  highest  amount  reached  or  desired 
at  the  top.  Men  representing  the  other  cities  could  be 
standing  around,  evidently  having  struck  the  machine, 
and  their  records  could  be  shown  on  a  bulletin  board  in 
the  background.  Another  man,  representing  the  home 
city,  could  be  shown  as  just  getting  ready  to  strike  the 
machine  to  see  what  he  can  do,  in  the  midst  of  words  of 
encouragement  or  taunts  from  the  other  men.  His  old 
record  might  appear  on  the  bulletin  board.  Underneath 
might  be  some  such  question  as,  "  Can't  he  send  it  to  the 
top?  "  "  Who  is  the  best  man?  "  "  How  much  will  he 
beat  his  old  record?  " 

2.    To  Show  Relative  Investments  in  School  Property 

In  the  Educational  Survey  of  Three  Counties  of  Alabama, 
the  ryurnber  of  dollars  invested  for  each  child  of  school 
age  by  each  state  is  given.2  Each  dollar  is  represented  by 
a  dollar  mark.  Thus,  Massachusetts  has  $115  invested 
in  school  property  for  each   child   of  school   age,   whil 

1  Suggested  by  Mr.  F.  ('.  Lowry  2  Page  212 


290  School  Statistics  and  Publicity 

Mississippi  has  only  $4.  The  advantages  of  this  graph 
are :  The  symbol  aids  in  calling  attention  to  the  graph ; 
the  length  of  the  row  of  dollar  marks  gives  the  effect  of  a 
bar ;  the  data  are  accurately  represented,  —  there  is  no 
material  exaggeration  or  anything  in  the  device  to  mis- 
lead. 

3.    To  Show  a  Lack  of  Funds  for  Maintenance 

In  the  Survey  of  Three  Counties  of  Alabama,1  there 
appear  pictures  of  a  schoolhouse  and  an  automobile. 
From  suitable  figures,  we  learn  that  the  initial  cost  of  a 
cheap  automobile  is  more  than  that  of  the  average  rural 
schoolhouse ;  and  that  the  upkeep  of  the  machine  is 
more  than  that  of  the  average  rural  school.  This  com- 
parison depends  for  its  power  on  contrast,  not  on  accuracy, 
for  there  is  nothing  particularly  accurate  about  it.  All 
the  same,  it  is  a  very  powerful  device  in  shaming  rural 
people  into  doing  their  duty  by  schools.  If  there  is  a 
single  automobile  in  such  a  district,  it  represents  more 
than  the  rural  school  expenses.  This  illustration  will 
be  of  service  chiefly  in  suggesting  similar  comparisons. 

The  Columbus  Dispatch  some  months  ago  had  a  very 
effective  cartoon  to  represent  the  disparity  in  wages  of 
women  teachers  and  statehouse  janitors  in  Ohio.2  It 
depicts  a  gruff  old  man  with  hands  in  his  pockets,  labeled 
"  Old  Man  Ohio."  Above  him  are  two  inserts.  The 
left  insert  represents  a  pitiful  woman  teacher  in  her 
classroom,  with  the  statement  that  the  average  salary 
for  the  public  school  teacher  in  Ohio  is  $54  a  month.  The 
right  insert  depicts  a  shuffling  negro  janitor  in  cap  and 
overalls,    bearing   broom,    mop,    and    bucket,    with    the 

1  Page  72 

2  Reproduced  in  American  School  Board  Journal,  Jan.,  1917,  p.  33 


Graphic  Presentations  of  School  Statistics    291 

statement  that  Ohio  pays  the  janitors  in  the  statehouse 
$60.  On  either  side  of  Old  Man  Ohio  is  a  hand  with  index 
finger  pointing  at  him  to  emphasize  the  title  of  the  whole 
—  "  For  Shame  !  " 


4.    To  Show  Length  of  School  Term,   Average  Attend- 
ance, Etc. 

For  this  the  Ayres  bulletin  on  the  forty-eight  states  has 
a  good  graph.  Each  day  is  represented  by  a  small  square, 
the  whole  representing  a  bar  graph,  with  each  bar  two 
squares  wide  to  make  the  bars  shorter.  The  total  length 
of  the  bar  represents  the  average  number  of  days  in 
which  schools  were  open  in  that  state.  Beginning  at  the 
left,  a  sufficient  number  of  these  little  squares  are  shaded 


48.NEW    MEXICO 


Fig.  7~>. —  Graph  for  Showing  the  Relation  of  the  Average  Number  of  Days' 
Attendance  by  Each  Pupil  to  the  Number  of  Days  School  Was  Open. 

(From  Dr.  Ayres's  Comparative  Study  of  the  Public  School  Systems  in  the  Forty-eight 
States.) 

to  represent  the  average  number  of  days  attended  by  each 
pupil  enrolled  in  that  state.  The  names  of  the  states 
appear  on  the  left  from  high  to  low,  beginning  with  Rhode 
Island,  which  had  her  school  open  193  days  and  kept 
each  pupil  in  148.8  days.  The  lowest  is  New  Mexico, 
which  had  her  schools  open  only  100  days  and  kept  each 
pupil  in  only  66.4  days.  This  is  represented  in  Figure  75. 
This  chart  shows  clearly  which  states  have  the  longest 
school  terms  and  which  are  making  the  best  use  of  what 
they  have.  It  could  be  used  for  cities  just  as  well  as 
for  states. 


292  School  Statistics  and  Publicity 

Another  way  to  show  the  attendance  of  different  school 
systems  is  suggested  by  the  graph  on  page  10  of  the  same 
bulletin.  This  shows  the  number  of  days  of  schooling 
each  child  of  school  age  would  get  per  year  if  he  got  his 
share.  Each  day  is  represented  by  a  small  dot ;  the 
dots  are  clustered  in  groups  of  five,  thereby  giving  a 
unit  of  the  week  as  well  as  the  day.  The  bar  effect  is 
obtained,  and  the  graph  has  every  advantage  mentioned 
for  those  above.  This  graph  could  be  used  in  any  graph- 
ing of  attendance,  probably.  The  copy  used  by  Ayres 
has  the  figures  at  the  right  end  of  the  bars,  which  is  bad, 
because  it  makes  the  bars  appear  lengthened  unequally. 
This  is  corrected  in  the  copy  below. 

48.   New  Mexico  46  : • : : • :  : • :  : • :  :• :  etc. 

5.    To  Show  Per  Capita  Costs  of  Schooling 

A  graph  for  this,  similar  to  the  last  two  described,  is 
found  on  page  18  of  the  Ayres  pamphlet.  It  sets  forth 
the  cost  of  one  day's  schooling  for  one  child  in  each  state 
in  1910.  Each  cent  is  represented  by  a  black  dot,  and  the 
bar  effect  is  obtained.  This  dot  or  the  cent  mark  or  the 
dollar  mark  could  be  used  in  graphing  any  data  on  costs, 
the  unit  being  chosen  so  as  to  keep  the  bar  short  enough.1 

South  Carolina  7  :  • :  :  • :  :  • :  :  • :  :  • :  etc. 

6.    To   Show  a  Disgraceful   State   of  Affairs  in   Certain 

Localities 

On  page  160  of  the  Alabama  three-county  survey  is 
shown  a  familiar  map  graph.  This  particular  one  shows 
the  map  of  the  United  States  with  all  states  having  com- 

1  Sec  page  260  of  this  book 


Graphic  Presentations  of  School  Statistics    293 

pulsory  education  laws  in  white,  and  those  not  having 
such  laws  in  black.  As  black  is  usually  associated  with 
shame  and  disgrace,  the  graph  becomes  a  stinging  accuser 
against  the  sections  that  are  backward  in  this  respect. 
The  same  idea,  of  course,  has  been  used  in  religious  maps. 
This  use  of  black  was  referred  to  in  the  description  of  the 
chart  on  page  32  of  the  Ayres  pamphlet.  (See  page 
244  of  this  book.)  The  idea  is  capable  of  wide  use  in 
cases  where  it  is  desirable  to  shame  backward  school 
systems  into  doing  something  better.  The  objection  that 
it  is  difficult  to  show  lettering  on  black  areas  is  easily  over- 
come by  using  white  ink  for  lettering. 

7.  To  Show  the  Variability  of  Children  in  the  Different 
Grades  in  Their  Achievement  in  Some  Standard  Test  or 

Similar  Matter 

A  discrete  distribution  or  Bobbitt  table  may  be  shown 
with  bar  graphs  as  shown  on  page  103,  with  a  curve  as 
shown  on  page  104,  or  with  a  scale  as  shown  on  pages  101 
and  102. 

For  a  continuous  distribution,  some  form  of  the  block 
graph  is  probably  best.  It  is  much  used  in  the  surveys  of 
Butte,  Salt  Lake  City,  and  other  places.  (See  pages  112 
and  137  of  this  book.) 

8.  To  Show  the  Relation  between  the  Rank  of  Children 
in  Their  Classes  in  the  Elementary   School  and  Their 

Probable  Entrance  into  High  School 

The  pupils  are  divided  into  three  classes,  the  upper, 
middle,  and  lower  thirds.  Each  third  is  represented  by  a 
broad  vertical  bar,  all  bars  the  same  length.     The  per- 


294 


School  Statistics  and  Publicity 


centage  of  each  bar  representing  those  not  going  on  should 
be  colored  black  or  shaded,  the  rest  being  left  unshaded. 
The  three  bars  should  be  placed  side  by  side,  with  the 
low  third  on  the  left  and  the  high  third  on  the  right.  (See 
Figure  76.) 


Low      Middle    High 
third        third     third 


Fig.  76.  —  Graph  Showing  Percentage  of  Eighth  Grade  Pupils  Entering 
High  School  from  the  Low  Third,  the  Middle  Third,  and  the  High  Third 
of  their  Classes,  Cleveland. 

(From  Cleveland  Survey,  Summary  Volume,  page  ISo,  by  permission.) 


9.    To  Compare  Achievements  of  the  Various  Grades  in 
Standard  Tests  with  Similar  Grades  from   Other    Cities 

The  best  graph  for  this  purpose  is  probably  a  modifi- 
cation of  the  bar  given  in  Figure  63  on  page  267.  Each 
grade  could  be  represented  by  the  one  kind  of  shading 
throughout.  Each  group  could  have  the  name  of  its 
city  beneath,  the  scale  could  appear  on  each  margin,  and, 
if  necessary,  faint  horizontal  scale  lines  could  be  drawn 
clear  across  the  drawing. 


Graphic  Presentations  of  School  Statistics    295 

10.    To  Show  Ratio  of  Lighting  to  Floor  Space 

In  the  Springfield  Survey,  a  graph  is  used  for  this  in 
which  the  floor  space  is  represented  by  a  black  square.1 
In  the  midst  of  this  is  a  white  square  representing  the 
lighting  space.  In  the  first  diagram  to  the  left  appears 
the  standard  ratio  ;  the  second  one  gives  the  average  ratio 
for  Springfield.  The  percentages  appearing  in  the  white 
squares  should  be  written  below.  White  squares  always 
appear  larger  than  black  ones  in  such  a  graph,  but  this 
one  is  too  small  for  the  feature  to  affect  it  materially. 
The  white  square  in  the  midst  of  the  black  gives  a  window- 
like effect  and  so  helps  to  call  attention  to  the  diagram. 
This  may  be  improved  upon  by  making  the  whole  the 
shape  of  a  side  wall,  with  the  white  the  shape  of  windows 
in  that  wall.     (See  Figure  72,  page  286.) 

The  preceding  illustrations  comprise  only  a  few  of  the 
best  selections  from  school  reports  and  surveys,  in  addition 
to  those  given  before.  It  will  be  noticed  that  many  of 
the  problems  the  superintendent  is  sure  to  meet  in 
graphing  his  data  have  not  been  mentioned.  The  bar 
graph,  concerning  which  much  has  been  said  in  the  pre- 
vious pages,  is  capable  of  wide  adaptation,  as  is  also  the 
Bobbitt  table.  One  or  the  other  of  these  two  may  be 
pressed  into  service  upon  almost  any  occasion.  It  is 
well,  however,  to  introduce  some  of  the  special  forms 
mentioned  above,  for  variety's  sake  if  nothing  else. 
The  efficient  superintendent,  of  course,  will  always  be  on 
the  lookout  for  improving  the  devices  we  now  have  for 
presenting  statistical  data  to  the  public,  or  he  may  work 
out   some   entirely   new   methods.     But   any   graph   he 

1  Page  24 


296  School  Statistics  and  Publicity 

devises  for  the  public  should  be  relatively  simple,  very 
clear,  and,  if  possible,  forceful. 

V.    ECONOMIES    IN    MAKING    SCHOOL    GRAPHS    FOR    THE 

PUBLIC 

Large  Cross-Section  Paper.  If  large  charts  are  to  be 
made  for  display  purposes,  excellent  results  may  be 
obtained  from  the  use  of  good  wax  or  grease  crayons  of 
assorted  colors.  The  time  required  for  getting  accurate 
measurements  may  be  greatly  reduced  by  the  use  of 
large  size  cross-section  paper,  as  the  counting  is  then  very 
easily  done  for  the  two  scales  or  for  locating  any  particular 
point.  The  paper  for  this  should  be  heavy  manila, 
light  enough  in  color  and  sufficiently  free  from  spots  to 
make  a  good  background  for  the  colors,  and  rough  enough 
to  take  the  colors  easily.  Sheets  36  by  40  inches  ruled 
faintly  in  one-inch  squares  give  excellent  results. 

These  may  be  obtained  from  any  large  printing  house  equipped 
with  ruling-pen  machines,  but  are  very  expensive  in  small  quantities. 
The  University  of  Chicago  Press  carries  them  in  stock  at  from  three 
to  five  cents  per  sheet,  depending  upon  the  price  of  paper,  transporta- 
tion extra ;  or  they  may  be  obtained  from  the  Peabody  College  Rook 
Store  on  the  same  terms. 

Making  Cartoons.  In  some  cases,  good  results  may 
be  obtained  by  pasting  pictures  on  charts,  if  the  cartoon 
effect  is  desired. 

For  example,  the  writer  wished  to  reproduce  in  large  form  the 
automobile  and  schoolhouse  graph  from  the  Alabama  three-county 
survey,  referred  to  on  page1  290.  He  got  a  picture  of  an  automobile 
from  a  large-sized  advertisement  in  the  Saturday  Evening  Pout  and  a 
picture  of  ;i  rural  schoolhouse  from  the  front  cover  of  an  American 
School  Hoard  Journal.  By  putting  on  the  title  and  the  figures,  the 
chart  was  soon  made.  Students  often  employ  this  same  device  in 
getting  up  posters  for  school  entertainments. 


Graphic  Presentations  of  School  Statistics    297 

Gummed  Letters.  Much  time  is  saved  and  a  beautiful 
chart  may  be  made  at  little  cost  of  time  and  money  by  the 
use  of  gummed  paper  letters  and  strips.  These  can  be 
gotten  in  several  colors  and  any  height  from  four  inches 
down.  The  superintendent  can  use  letters  from  one  to 
one  and  a  half  inches  high,  costing  from  about  $1.50  to 
$2.00  per  thousand.  The  letters  may  be  spaced  very 
easily  and  quickly  if  the  chart  is  made  on  a  large  sheet 
of  cross-section  paper. 

These  letters  and  strips  may  be  obtained  from  the  Tablet  and 
Ticket  Company  of  Chicago,  which  will  send  pictures  of  such  charts. 
The  strips  are  very  serviceable  for  making  bar  graphs. 

Rubber  Stamp  Set.  An  advertising  set  such  as  is  used 
by  merchants  for  display  cards  can  be  used  advantageously 
in  making  good  charts.  With  a  little  practice,  neat 
charts  can  be  quickly  made  with  such  a  stamp  set, 
especially  on  the  large  cross-section  sheets. 

A  satisfactory  set  may  be  obtained  from  the  Milton  Bradley 
Company,  73  Fifth  Avenue,  New  York  City,  or  from  Salisbury- 
Schulz  Company,  157  West  Randolph  Street,  Chicago,  or  at  any  of 
their  various  offices,  for  about  $5,  depending  upon  the  price  of 
rubber. 

Securing  Clear  Lines.  In  drawing  charts  for  cuts, 
the  chart  and  all  essential  cross-section  lines  should  be 
reproduced.  The  error  of  reproducing  the  background 
of  numerous  cross-section  lines  may  be  avoided  by  tracing 
over  in  India  ink  the  parts  that  should  come  out  in  the 
cut.  The  photographic  process  will  reproduce  this  easily 
before  the  background  with  its  fainter  tones  will  "  take." 
The  too  prominent  background  is  usually  due  to  making 
the  drawings  with  ordinary  ink  or  faint  typewriter  letter- 


298 


School  Statistics  and  Publicity 


ing.  The  latter  may  be  easily  traced  in  India  ink  and 
give  good  results.     (See  page  76.) 

Miscellaneous  Aids.  Numerous  aids  to  making  charts 
quickly  with  various  kinds  of  cross-section  paper,  special 
scales,  etc.,  may  be  obtained  from  the  Educational 
Exhibition  Company,  Providence,  Rhode  Island,  or  the 
Tablet  and  Ticket  Company  of  Chicago. 

"  Perpetual  "  Attendance  Graph  Device.  An  exceed- 
ingly easily  managed  graph  arrangement  was  observed 


ATTENDANCE 

[2  a*  [Month 


1A   IB  1C  JVB.IIA  llBlllAMBIVA  V    VI  VII  V 111 


Fig. 


Perpetual"  Attendance  Graph  Device. 


some  years  ago  by  the  writer  in  the  device  used  by 
Principal  R.  L.  Dimmitt  of  the  Ensley  High  School  at 
Birmingham,  Alabama,  to  compare  the  attendance 
record  of  the  classes  in  the  high  school.     (See  Figure  77.) 


Graphic  Presentations  of  School  Statistics    299 

A  large  chart  was  made  up  on  Bristol  board,  once  for  all,  with  the 
percentage  scale  running  up  on  the  sides.  Each  class  was  repre- 
sented by  a  paper  ribbon  that  came  through  a  slit  on  the  base  line. 
By  pulling  the  ribbons  up  and  down  each  month  and  fastening  the 
ends  with  thumb  tacks,  the  graph  was  quickly  brought  up  to  date. 
The  omission  of  the  zero  line  exaggerated  differences,  but  such  ex- 
aggeration was  to  some  extent  desired  for  emphasis.  The  only  real 
drawback  seems  to  be  that  one  cannot  compare  the  records  of  the 
classes  by  month  with  such  a  chart.  The  chart  as  operated,  however, 
does  not  need  to  show  comparisons.  The  emphasis  on  attendance 
is  intended  to  keep  up  attendance  all  the  time,  and  not  to  let  children 
slack  up  one  month  because  of  a  good  record  the  preceding  month. 
If,  however,  it  is  desirable  to  make  comparisons,  this  can  be  easily 
done  by  preserving  kodak  pictures  of  the  chart  at  various  times. 
This  cut  is  drawn  from  such  a  picture. 

Graphs  without  Cuts.  If  the  charts  are  to  be  set  up 
in  type  and  not  with  cuts,  the  originals  may  be  made 
either  by.  hand  or  on  the  typewriter.  For  all  such  work 
the  horizontal  or  vertical  bar  graph  is  especially  useful. 
By  the  aid  of  conventional  signs,  different  lengths  of 
rules,  etc.,  almost  any  chart  containing  a  bar  effect  can 
be  made  so  simple  that  it  can  be  set  up  in  any  ordinary 
limitedly  equipped  printing  office.  The  following  are 
examples : 


xx 
xxxx 
xxxxxx 
########  xxxxxxxxx 

xxxxxxx  xxxxxxxxxxxx 

0000     i  xxxxxxxxxxxxxxxx 

------_---------_----     xxxxxxxxxxxxxxxx 


Utilizing  Students.     In  most  of  the  work  on  graphs  for 
the  public,  the  superintendent  need  only  furnish  the  idea 


300 


School  Statistics  and  Publicity 


or  design  for  the  chart.  The  rest  of  the  work  can  be 
done  by  various  upper  grades  or  high  school  classes  as 
very  profitable  laboratory  or  practical  exercises.  The 
mathematics  classes,  especially  those  in  algebra,  and  the 
drawing  and  art  classes  are  the  logical  ones  to  call  upon 
to  take  charge  of  this  work. 

In  Newton,  Massachusetts,  the  boys  in  the  high  school  printed 
the  title  pages  of  the  annual  school  reports  and  probably  made  the 
graphs,  although  this  latter  is  not  so  stated.  In  the  World  Book 
Company's  reprint  of  MacAndrew's  The  Public  and  Its  School,  the 
drawings  are  all  made  by  public  school  children.  Children  who  could 
do  such  drawing  could  make  most  of  the  graphs  advocated  in  this 
book.     As  it  is,  practically  all  the  drawings  in  this  book  have  been 


TtJCWT  OF  HjPILS    ATTA1/YIIVG    01  VEK  SCOS;fS    (K   5TGKE    RMSWIN'G 
rESTS,    BDTH-  SURVEY,    p.    9;,.       ^\)    pup, |$  ^estej  ) 


Fig.  78. 


showing  Sketch  and   Rough  No1< 
Figure    13. 


Which  a  Pupil   Drew 


copied  or  drawn  from  the  author's  suggestions  and  statistical  data 
by  the  students  of  Mr.  E.  S.  Maclin  at  the  Atlanta  Technological 
High  School,  as  a  demonstration.  Figure  78  shows  the  sketch  and 
notes  furnished  one  of  these  pupils  by  the  writer,  from  which  Figure 
13  was  drawn.  Figure  79  shows  the  cartoon  drawn  by  a  student 
from  notes  given  by  Mr.  Maclin. 


Graphic  Presentations  of  School  Statistics    301 


ONLY  TOO  TRUE 


Fig.  79.  —  Cartoon  Drawn  by  an  Atlanta  High  School  Student. 

This  cartoon  represents  the  high  school  situation  in  Atlanta  and  was  drawn  by  the 
student  with  only  these  suggestions  from  his  drawing  instructor,  Mr.  E.  S.  Maclin: 
Subject:  The  overcrowded  conditions  of  the  Atlanta   High  Schools.      Represent  a 
good-natured  boy  who  has  outgrown  his  clothes  at  every  point,  trousers  splitting, 
feet  running  out  of  his  shoes,  shirt  too  small  for  him,  etc.      He  is  represented  as 
saying  to  his  father,  the  City  Fathers,  "Dad,  I'm  growing  in  spite  of  you.      I'll 
soon/be  in  the  street  and  no  place  to  go."      His  father  is  represented  as  being  a 
rich   man   with    Atlanta's  per   capita  bonded   indebtedness  of  about  $22   per  in- 
habitant. 
It  will  be  noted  that  the  suggestions  were  not  wholly  followed,  but  the  cartoon  aa 
published  in  an  Atlanta  paper  was  sufficiently  forceful. 


302  School  Statistics  and  Publicity 

EXERCISE 

Take  the  school  report  or  school  survey  used  in  the  exercise  on 
page  233.  Write  out  a  detailed  criticism  of  the  graphs  or  lack  of 
them  in  it  from  the  standpoint  of  their  effectiveness  with  the  public, 
showing  just  why  they  are  good  or  liable  to  be  unsuccessful.  In  the 
cases  of  the  unsuccessful  ones,  or  failure  to  use  graphs  where  desirable, 
sketch  graphs  that  would  present  the  same  data  properly. 

REFERENCES  FOR  SUPPLEMENTARY  READING 

Brinton,  W.  C.  Graphic  Methods  for  Presenting  School  Facts.  Prac- 
tically all. 

Ellis,  A.  Caswell.  "The  Money  Value  of  Education."  U.  S.  Bureau 
of  Education  Bulletin,  1917,  No.  22. 

King,  W.  I.     Elements  of  Statistical  Method,  Chapter  X. 

Rugg,  H.  O.     Statistical  Methods  Applied  to  Education,  Chapter  X. 


CHAPTER   XII 

TRANSLATING    STATISTICAL    MATERIAL    ON 
SCHOOLS    FOR   THE   PUBLIC 

I.    THE    NEED    OF   TRANSLATION 

For  the  superintendent  who  wishes  to  present  his  school 
statistics  effectively  to  the  public,  three  devices  are 
available.  He  may  graph  his  material ;  he  may  tabulate 
it ;  he  may  translate  it  into  words.  Each  procedure  has 
its  strengths  and  weaknesses.  Each  is  best  adapted  to 
certain  conditions. 

Translation  is  the  most  serviceable  device  in  cases 
where  it  is  difficult  to  secure  or  to  have  printed  good 
tabulations  or  graphs.  A  translation  of  statistical 
material  into  words  with  a  few  mere  numbers  can  be 
typewritten  or  set  up  in  type  anywhere  with  little  effort. 
Such  a  translation  can  be  read  or  spoken  at  any  public 
meeting  with  no  particular  preparation.  Furthermore, 
it  can  be  so  neatly  expressed  that  persons  hearing  it  can 
easily  remember  it  and  quickly  pass  it  on  to  others,  a 
thing  not  possible  with  tabulations  and  graphs.  It  can 
be  so  forcibly  worded  that  it  will  arouse  people  to  action. 

For  example,  the  New  York  Survey  devoted  many 
pages  to  figures  and  graphs  setting  forth  the  results 
attained  in  arithmetic.  But  it  is  very  doubtful  if  the 
whole  or  any  part  of  this  section  could  impress  the 
average  man  as  does  the  simple  translation  of  the  summary 

303 


304  School  Statistics  and  Publicity 

by  Mc Andrew :    "It  takes  us  less  time  to  get  a  thing 
wrong  here  than  it  does  in  the  average  school  system."  l 

Statistical  material  on  schools  is  often  presented  in 
words  in  such  a  way  as  to  be  clear  to  the  trained  school  man 
and  yet  be  unintelligible  to  the  average  man.  If  it  is  to 
be  clear  to  the  latter,  it  must  be  as  definitely  translated 
for  him  as  must  material  from  the  Latin  or  other  foreign 
languages,  from  a  doctor's  description  in  medical  parlance 
of  a  disease,  from  fundamental  political  theories,  or  from 
scientific  experiments  in  agriculture.  We  have  various 
writers  to  translate  the  classics ;  we  have  Woods  Hutchin- 
son and  other  medical  writers  to  translate  medical 
knowledge ;  we  had  Miinsterberg  and  James  to  translate 
psychology ;  we  have  various  writers  and  speakers  to 
translate  political  theories  into  the  party  campaign 
books ;  the  agricultural  colleges  have  numerous  writers 
of  bulletins  to  translate  agricultural  knowledge  for  the 
farmer  and  housewife.  Have  we  not  just  as  much  need 
for  translating  school  statistics  into  the  language  of  the 
average  man? 

Beyond  clearness,  the  matter  of  force  is  very  important. 
The  translation  must  not  only  be  clear  to  the  man  for 
whom  it  is  intended,  but  it  must  take  hold  of  him  in  some 
way.  Force  cannot  be  obtained  by  the  mere  repetition  of 
tabulations  in  straight  sentences  of  reading  matter,  how- 
ever much  some  state  superintendents  appear  to  think 
it  can.  This  is  not  translation.  The  problem  is  really 
the  same  problem  as  that  of  the  life  insurance  compa- 
nies, the  corporation  seeking  to  influence  the  public,  and 
advertisers  in  general,  when  they  try  to  reach  the  public 
with  statistical  material.  They  do  far  more  than  merely 
express  figures  by  words. 

1  The  Public  and  Its  Schools,  p.  8 


Translating  Statistical  Material  305 

It  cannot  be  too  strongly  stated  that  translation  is 
not  the  same  as  definition  or  explanation.  In  defining 
statistical  terms  we  simply  aim  to  show  precisely  what 
we  mean  by  them,  often  in  terms  just  as  technical.  In 
explaining  statistical  terms  we  merely  try  to  make  our 
particular  use  of  them  clear  to  persons  who  already  have 
the  same  definitions  of  them  as  ourselves,  this  too  in 
language  often  just  as  technical.  Good  translation  of  a 
statistical  term,  of  course,  involves  both  definition  and 
explanation,  but  it  is  more.  It  puts  the  emphasis  on  the 
fact  that  the  meaning  must  be  carried  over  into  an  entirely 
different  language  or  set  of  expressions.  For  example,  the 
median  may  be  defined  as  the  magnitude  of  the  midpoint 
in  a  distribution.  In  any  given  frequency  table  or  sur- 
face of  frequency,  it  may  be  explained  that  the  point 
marked  "  m  "  signifies  the  median.  But  the  idea  of  the 
median  in  a  set  of  superintendents'  salaries  may  be 
translated  for  the  average  man  by  telling  him  that  half 
the  superintendents  get  more  than  that  salary  and  half 
of  them  get  less. 

The  writer  has  never  seen  any  discussion  of  this  trans- 
lation phase  of  school  statistics.  Consequently,  the 
treatment  here  is  only  a  preliminary  or  tentative  analysis, 
to  be  supplemented  with  illustrations  from  any  source. 
For  clearness  and  brevity,  the  points  will  be  given  rather 
dogmatically. 

,     II.    SUGGESTIONS   FOR   GOOD    TRANSLATIONS 

The  main  things  to  be  kept  in  mind  in  working  out 
good  translations  of  school  statistics  are  : 

1.  The  illustrations  and  images  used  must  be  of  an 
elementary  nature,  or  at  least  familiar  to  the  people  for 
whom  the  translation  is  being  made. 


306  School  Statistics  and  Publicity 

The  announcement  that  retarded  children  in  school  are  as  thick  as 
Ford  automobiles  or  men  wearing  Masonic  pins  is  intelligible  any- 
where. On  the  other  hand,  the  statement  that  retarded  children  are 
as  thick  as  negroes  in  southern  cities  would  be  a  very  effective  way  of 
translating  the  facts  to  some  southern  audiences,  but  would  hardly 
be  of  much  value  elsewhere.  It  would  not  even  work  in  all  southern 
cities,  because  the  proportion  of  negroes  in  them  varies  from  close  to 
50  per  cent  down  to  less  than  5  per  cent. 

2.  In  some  cases  it  may  be  necessary  to  use  several 
illustrations  in  order  to  be  sure  of  reaching  all  classes  of 
people. 

3.  Instead  of  representing  a  total  by  imagining  an 
unreal  extension  of  a  familiar  object,  or  by  making  up 
from  familiar  units  an  aggregate  so  large  as  to  be  in- 
comprehensible, it  is  usually  better  to  employ  some  other 
unit.     Often  this  other  unit  is  one  of  time. 

It  is  of  doubtful  value  to  ask  the  average  man  to  think  of  a  line 
of  school  children  eight  hundred  miles  long;  of  a  schoolhouse  as  large 
as  all  the  schoolhouses  of  the  county  put  together;  of  a  sheet  of 
writing  paper  large  enough  to  cover  a  township ;  of  a  lump  of  coal 
weighing  as  much  as  all  the  coal  burned  in  one  day  in  many  schools ; 
or  of  a  total  of  any  sort  reaching  into  the  hundreds  of  thousands. 
Such  translations  are  sometimes  attempted.  They  might  well  be 
called  Jack-and-the-beanstalk  translations,  for  they  are  about  as  far- 
fetched. 

Practically  all  statistical  totals  needing  to  be  translated  will,  on 
examination,  be  found  to  involve  in  some  way  units  of  length,  area, 
volume,  weight,  and  time.  The  aim  should  be  to  translate  the  total 
into  another  kind  of  unit  that  will  keep  it  within  the  limits  of  com- 
prehension or  experience  of  the  ordinary  man.1  Often  a  relatively 
larger  unit  of  time  that  is  forceful  may  be  employed  than  in  the  case 
of  the  other  kinds  of  units,  because  most  people  comprehend  long 
stretches  of  time  or  the  consumption   of  goods  over  long  periods, 

1  For  elaborating  this  point  the  writer  is  indebted  to  Mr.  H.  A. 
Webb,  one  of  his  students. 


Translating  Statistical  Material  307 

fairly  well.     The  periods  of  time,  however,  must  in  general  be  well 
within  the  limits  of  the  ordinary  man's  active  life  span. 

Consider  the  following  translation  of  the  average  daily  absence 
for  the  schools  in  Texas  in  1911-13  : 

"Placed  twelve  feet  apart,  these  white  pupils  absent  every  day 
from  the  schools  of  Texas  would  form  a  line  extending  across  the 
state  from  El  Paso  to  Texarkana,  a  distance  of  over  eight  hundred 
miles."  1 

The  average  Texan,  even  though  he  is  from  an  early  age  accustomed 
to  boasting  of  the  size  of  the  state,  can  have  but  a  very  hazy  idea  of 
the  distance  mentioned.  Even  if  he  has  traveled  from  one  city  to 
the  other,  his  idea  is  dependent  chiefly  upon  his  recollection  of  the 
time  it  took  to  make  the  trip,  and  a  good  part  of  this  time  he  may  have 
been  asleep.  In  all  likelihood,  he  has  never  seen  children  lined  up 
twelve  feet  apart  this  way.  It  would  probably  be  better  for  most  people 
to  consider  these  children  as  marching  double  file  and  say  how  many 
days  it  would  take  them  to  pass  a  given  point.  The  average  man  has 
seen  people  marching  double  file  in  parades  and  has  a  good  idea  of 
about  how  fast  they  would  pass. 

Instead  of  saying  that,  if  the  school  costs  of  a  certain  city  were 
represented  by  silver  dollars  lying  side  by  side,  they  would  extend 
the  distance  of  ten  blocks,  it  would  be  far  better  to  say  that  at  a  cer- 
tain sum,  say  $50  a  month,  it  would  take  a  man  a  certain  number  of 
years  to  earn  an  equivalent  amount  of  money.  In  this  case,  the 
average  man  has  a  much  better  idea  of  how  long  it  takes  him  to  earn 
money  than  he  has  of  the  distance  dollars  touching  each  other  will 
extend.  He  has  never  seen  silver  dollars  lined  up  along  a  street,  but 
he  has  worked  for  wages  and  knows  the  value  of  his  money. 

A  very  effective  example  of  this  sort  is  found  in  the  Negro  Year- 
Book  for  1916-17?  The  compiler  of  this  made  a  study  of  the  average 
number  of  days  that  each  negro  child  of  school  age  attended  school 
in  each  of  the  southern  states.  To  make  the  smallness  impressive,  he 
calculated  the  number  of  years  it  would  take  the  average  negro  child 
to  complete  the  elementary  school  on  the  basis  of  eight  grades  and 
nine  months  to  the  school  year,  thus  : 

1  White,  E.  V.:  "A  Study  of  Rural  Schools  in  Texas,"  Bulletin 
of  University  of  Texas,  No.  364,  Oct.  10,  1914,  p.  20 

2  Page  233 


308  School  Statistics  and  Publicity 

No.  of  yrs.  it  would  take 
the  average  negro  child  to 
complete  the  elementary 
course  in  the  public  schools 
State  provided  for  him 

Maryland  16 

Texas  18 

Virginia  18 

Georgia  19 

Florida  20 

North  Carolina  20 

Alabama  22 

Louisiana  25 

South  Carolina  33 

This  could  have  been  made  more  forcible  by  changing  the  figures  to 
the  age  at  which  a  negro  entering  school,  when  six  years  old,  could 
complete  the  grades,  as  22  for  Maryland,  24  for  Texas,  etc. 

4.  In  cost  statistics,  it  is  sometimes  advisable  to 
minimize  the  total  by  expressing  it  in  amount  per  small 
unit  of  time,  usually  a  trivial  sum. 

Thus  daily  papers  and  many  weekly  periodicals  advertise  "  10  cents 
a  week"  and  do  not  call  attention  to  the  total  of  $5.20  for  the  year. 

A  church  calls  for  "30  cents  a  week"  and  does  not  emphasize  the 
fact  that  this  reaches  $15.60  for  the  year. 

The  Y.  M.  C.  A.  announces  that  membership  with  all  privileges 
costs  "a  nickel  a  day  or  the  price  of  your  daily  cigar,  "  when  the  yearly 
total  is  from  $15  to  $18,  without  mentioning  these  latter  figures. 

The  Liberty  Loan  called  for  "a  dollar  a  week"  instead  of  stressing 
the  $50  that  would  be  laid  aside  for  the  year. 

The  sum  for  the  total  is  generally  not  stressed  except  in  savings 
bank  advertisements  and  such  appeals,  where  the  aim  is  to  surprise 
the  reader  with  the  total  saved,  and  not  any  amount  that  is  expended. 

5.  Absolute  accuracy  frequently  has  to  be  sacrificed 
to  force  and  clearness  in  translation. 

Thus,  if  the  percentage  of  negroes  in  a  city  population  was  40,  and 
the  percentage  of  retarded  children  35  or  45,  the  negro  illustration 


Translating  Statistical  Material  309 

mentioned   before   would   be   accurate   enough   for   translating   the 
idea. 

The  "nickel  a  day"  of  the  Y.  M.  C.  A.  is  adequate  for  any  an- 
nual sum  from  $15  to  $18. 

6.  Practically  all  totals  have  to  be  translated  through 
comparisons,  using  familiar  objects  or  notions,  before  they 
can  be  understood  or  have  much  force  for  the  average  man. 

The  statement  that  four  per  cent  of  all  children  of  school  age  are 
mentally  defective  and  need  special  attention  would  mean  little  to 
most  men.  But  to  say  that  every  school  system  enrolling  500  chil- 
dren had  among  that  number  20  defective  pupils,  or  enough  to  equal 
the  ordinary  6A  grade  taught  by  one  teacher,  would  drive  this  fact 
home  to  the  average  man. 

The  mere  statement  that  there  were  on  the  average  64  children  to 
each  grade  teacher  in  a  school  system  would  mean  little  to  the  people 
of  the  city  and  probably  would  receive  no  attention.  However,  if 
the  statement  were  changed  to  read  that  there  were  enough  surplus 
children  over  the  standard  number  for  the  various  classes  to  fill  four 
classrooms,  the  fact  would  certainly  impress  citizens. 

The  bare  announcement  of  the  amount  spent  for  some  item  of 
school  expense  is  not  nearly  so  forceful  as  the  statement  that  it  is 
only  a  certain  fraction  of  the  ice  cream  and  soda,  liquor  or  tobacco 
expenditures  of  the  town,  if  such  figures  or  approximate  estimates 
of  them  can  be  obtained  from  merchants. 

A  southern  county  superintendent  recently  translated  his  valua- 
tion estimates  as  follows : 

"The  total  value  of  all  school  property  in  the  county  outside 
of  the  city  is  $34,420,  which  is  $3080  less  than  half  the  value  of 
the  county  jail  and  site,  and  $3000  less  than  one  fourth  of  the 
value  of  the  county  courthouse,  site,  furniture  and  fixtures. 

'tOf  the  $34,420  invested  in  school  property  in  the  county,  only 
$15,665,  or  less  than  half,  belongs  to  the  state  and  county.  In 
other  words,  the  value  of  all  the  school  property  of  those  schools 
with  titles  vested  in  the  state  is  $5335  less  than  the  cost  of  three 
of  her  best  motor  trucks  used  in  constructing  good  roads. 

"The  total  value  of  all  supplies  and  equipment,  including 
musical  instruments  and  libraries,  is  $5875,  which  is  $1125  less 


310  School  Statistics  and  Publicity 

than  the  cost  of  one  motor  truck  used  in  building  the  roads  of 
the  county. 

"All  school   equipment   in   the   county  outside   the   city   of 

is  equal  in  value  to  less  than  one  seventh  of  the  value 

of  the  machinery  owned  by  the  county  and  used  in  the  making 
of  good  roads.  'Seven  to  one'  is  the  ratio  of  the  county's  in- 
vestment in  equipment  for  making  roads  as  compared  to  her  in- 
vestment in  equipment  for  making  men  and  ivomen." 

The  same  superintendent  translated  the  total  area  of  all  the  school 
grounds  in  the  county,  73  acres,  by  comparing  it  with  the  180  acres 
of  playgrounds  in  the  property  of  two  country  clubs  near  the  county- 
seat. 

In  similar  fashion,  the  increased  enrollment  of  the  public  high  schools 
of  the  United  States  in  1914  over  1913  would  make  a  city  as  large  as 
Chattanooga  and  Knoxville,  Tennessee,  combined. 

Dr.  L.  P.  Ayres  did  a  great  service  for  the  school  survey  movement 
when  he  used  the  following  translation  to  show  its  results : 

"About  seven  years  ago  this  retardation  became  one  of  the 
most  widely  studied  problems  of  educational  administration,  and 
in  the  past  four  it  has  been  one  of  the  prominent  parts  of  the 
school  survey.  During  the  entire  period,  hundreds  of  superin- 
tendents throughout  the  country  have  been  readjusting  the 
schools  to  better  the  conditions  disclosed. 

"In  these  seven  years  the  number  of  children  graduating  each 
year  from  the  elementary  schools  of  America  has  doubled.  The 
number  now  is  three  quarters  of  a  million  greater  annually  than 
it  was  then.  The  only  great  organized  industry  in  America  that 
has  increased  the  output  of  its  finished  product  as  rapidly  as  the 
public  schools  during  the  past  seven  years  is  the  automobile 
industry."  ' 

The  Warren  County  (Kentucky)  Bulletin  translates  the  losses 
through  retardation  as  amounting  to  "thousands  of  years."  This 
is  probably  more  impressive  than  really  comprehensible. 

Suppose  a  teacher  is  repeating  every  answer  after  every  pupil, 
hour  in  and  hour  out,  for  days,  as  did  some  teachers  observed  by  the 
author's  students.     To  tell  a  citizen  that  such  a  teacher  is  wasting 

1  Ayres,  L.  P.:  "A  Survey  of  Surveys,"  Indiana  University  Bul- 
letin, Vol.  13,  No.  11,  p.  180 


Translating  Statistical  Material  311 

time  would  moan  little.  But  to  show  him  that  if  the  teacher  had 
forty  pupils,  she  probably  would  have  twenty  of  them  reciting  half 
a  day  each  day;  that  if  she  repeated  every  answer,  she  could  only 
cover  about  half  as  much  as  if  she  did  not  repeat ;  that  this  meant 
virtually  wasting  a  fourth  of  a  day  for  every  pupil ;  that  a  fourth  of  a 
day  for  forty  pupils  amounted  to  ten  days  for  one ;  that  this  ten  days 
was  two  weeks  in  school  for  one  pupil ;  that  this  teacher  by  her  repe- 
titions was  each  day  consuming  or  wasting  the  equivalent  of  one 
pupil's  time  for  two  whole  weeks  in  school,  —  all  this  would  mean  a 
good  deal  to  him. 

7.  Many  questions  involving  value,  and  particularly 
exhibits  of  loss  or  waste,  can  be  profitably  translated 
into  a  money  equivalent.  This  is  particularly  true  of  all 
proposals  involving  an  increase  in  school  taxes,  which 
must,  of  course,  be  addressed  to  the  taxpayers. 

Thus,  the  need  of  health  education  and  sanitation  may  be  shown 
by  translating  the  loss  in  money  through  death  and  sickness  into  the 
cost  estimates  furnished  by  Professor  Irving  Fisher  of  Yale,  and 
others.  This  is  well  done  in  the  Warren  County  (Kentucky)  Survey, 
page  2.  It  was  found  there  that  in  the  year  1915  there  were  155 
preventable  deaths,  and  the  potential  loss  occasioned  by  these  deaths, 
according  to  Professor  Fisher's  estimate,  was  shown  to  be  $263,500. 

The  well-known  chart  of  the  United  States  Bureau  of  Education, 
which  attempted  to  show  that  every  day  spent  in  school  is  worth 
$9  to  a  child,  is  a  good  example.  The  fallacies  in  it  are  likely  to 
give  actual  pain  to  any  one  who  knows  much  about  statistical 
method,  or  who  will  use  his  common  sense  effectively.  But  it  does 
translate  the  material  into  something  that  is  intelligible  and  appealing 
to  most  people.  It  has  been  of  great  value  for  thousands  of  high 
school  commencement  addresses  and  campaigns  for  increasing  school 
levies. 

In  any  propaganda  for  increasing  school  taxes,  it  is  well  to  trans- 
late the  increase  into  the  respective  amounts  of  money  which  will  be 
due  from  men  who  already  pay  certain  round  sums  for  total  taxes, 
the  number  in  each  class,  thus : 

This  increase  in  school  taxes  will  require : 

50  cents  more  per  year  from  each  of  the  1500  men  who  now  pay  a 
total  yearly  tax  of  $5  each. 


312  School  Statistics  and  Publicity 

$1  more  per  year  from  each  of  the  500  men  who  now  pay  a  total 
yearly  tax  of  $1Q  each. 

$2  more  per  year  from  each  of  the  250  men  who  now  pay  a  total 
yearly  tax  of  $20  each,  etc.,  etc. 

III.    EXAMPLES  OF  GOOD  TRANSLATIONS  OF  SCHOOL 
STATISTICS 

In  many  of  the  school  surveys  which  use  modern  statis- 
tical method,  the  technical  statistical  terms  and  results 
have  not  been  translated  so  as  to  influence  the  average 
man.  To  just  this  extent  they  are  certain  to  fall  short 
of  what  a  survey  or  review  of  a  school  claims  to  be. 
True,  there  are  to  be  found  in  isolated  places  some  very 
successful  translations  of  these  terms  But  these  are  so 
scattered  as  not  to  be  generally  accessible.  The  remainder 
of  this  chapter  aims  to  make  a  few  of  these  available  for 

general  use. 

1.    Sampling 

The  following  translation  of  the  process  of  sampling 
was  used  by  the  author's  students  in  preparing  some 
material  for  a  survey  of  one  of  the  western  cities : 

In  giving  the  standard  tests  to  the  children  of  this  city,  it  was  found 
to  be  too  great  a  task  to  test  every  child,  as  labor  in  grading  the  papers 
would  be  enormous.  So  certain  schools  and  grades  were  chosen  at 
random,  and  tests  were  given  in  these  places. 

When  a  carload  of  wheat  is  being  graded,  the  grader  does  not 
look  at  all  the  grains,  as  every  one  knows  that  would  take  too  long. 
What  he  does  is  to  take  a  few  grains  from  each  of  several  places  well 
distributed  throughout  the  whole  lot  of  wheat,  and  to  make  his  rating 
from  these  samples.  This  process  has  been  found  accurate  enough  so 
that  it  is  constantly  used  in  business  without  complaint  from  either 
the  buyer  or  seller.  The  same  thing  is  true  in  grading  fruit.  Not 
every  apple  or  peach  is  actually  looked  at,  or  even  every  box,  but 
only  certain  apples  or  peaches  taken  from  certain  boxes  (determined 
at  random),  and  the  quality  of  these  determines  the  grade  assigned 
to  the  whole  consignment. 


Translating  Statistical  Material  313 

These  examples-  are  well  known  to  the  people  of  this 
section,  in  all  probability  because  of  the  amount  of  wheat 
and  fruit  raised  in  the  state.  After  they  have  read  such 
a  translation,  are  they  likely  to  doubt  the  validity  of  the 
sampling  done  in  the  school  survey? 

Sampling  may  also  be  explained  by  comparing  it  with 
the  process  of  taking  a  straw  vote.  Most  men  understand 
how  this  is  done,  how  reliable  it  is,  etc. 

2.  The  Average 

A  familiar  idea  for  translating  the  average  is  afforded 
by  the  "  see-saw  "  or  by  the  lever.  Every  one  knows  that 
the  farther  the  person  is  from  the  object  upon  which  the 
lever  is  resting,  the  more  weight  it  takes  on  the  other 
side  to  counterbalance  him.  That  is,  the  center  is  what 
the  physicist  means  by  "  center  of  gravity."  It  is  the 
same  with  the  average.  The  average  is  the  balancing 
point  of  all  the  cases  in  a  distribution,  with  their  distances 
from  it  and  their  sizes  taken  into  account.  The  farther 
an  item  is  from  the  average,  the  more  weight  it  has. 

Professor  King  uses  the  expression  "  a  type  "  for  the 
average. 

3.  The  Median 

The  following  translations  are  taken  from  surveys : 

"Among  the  teachers  in  the  elementary  schools,  the  median  or 
midway  age  is  twenty-nine  years,  half  of  the  teachers  being  twenty- 
nine  years  old  or  older,  and  the  other  half,  twenty-nine  years  of  age 
or  younger.    1 

"With  teachers  ranked  in  descending  order  according  to  size  of 
salaries,  the  median  salary  is  the  salary  received  by  the  teacher  half 
way  down  the  line."  2 

1  "The  Public  Schools  of  Springfield,  Illinois,"  Springfield  Survey, 
P-  59  2  "Financing  the  Schools,"  Cleveland  Survey,  p.  53 


314  School  Statistics  and  Publicity 

The  idea  of  a  median  was  translated  with  the  aid  of  a  picture  by 
Superintendent  Womack  in  his  Conway  (Arkansas)  Survey  by  taking 
his  ungraded  class  and  lining  them  up  for  height  from  low  to  high. 
The  middle  child  was  standing  on  a  drain  which  ran  out  white  in 
front,  and  there  were  two  gaps  in  the  line  to  indicate  the  quartiles. 
Any  reader  could  look  at  the  line  of  pupils  and  quickly  get  a  clear 
notion  of  what  was  meant  by  median  height. 

"  The  point  above  which  and  below  which  fifty  per  cent  of  the  cases 
fall."  » 

"  The  median  is  the  case  which  was  found  in  the  investigation  to 
have  as  many  cases  below  it  as  there  are  above  it."2 

A  very  effective  translation  of  the  median  can  be  se- 
cured by  using  a  description  of  a  typical  person  or  school, 
which  will  have  the  median  amount  in  each  of  a  number 
of  different  qualities.  The  first  example  of  this  use  of 
the  median  ever  noted  by  the  author  was  the  description 
of  the  typical  teacher  in  Professor  Coffman's  Social  Com- 
position of  the  Teaching  Population,  pages  79-80.  A  por- 
tion of  this  description  follows : 

The  typical  American  male  public  school  teacher  ...  is  twenty- 
nine  years  of  age,  having  begun  teaching  at  almost  twenty  years  of 
age,  after  he  had  received  but  three  or  four  years  of  training  beyond 
the  elementary  school.  In  the  nine  years  elapsing  between  the  age 
he  began  teaching  and  his  present  age,  he  has  had  seven  years  of  ex- 
perience, and  his  salary  at  the  present  time  is  $489  a  year.  Both  his 
parents  were  living  when  he  entered  and  both  spoke  the  English 
language.  They  had  an'  annual  income  from  their  farm  of  $700, 
which  they  were  compelled  to  use  to  support  themselves  and  their 
four  or  five  children. 

His  first  experience  as  a  teacher  was  secured  in  the  rural  schools, 
where  he  remained  for  two  years  at  a  salary  of  $390  per  year.     He 

1  Cubberley,  E.  P.:  "Survey  of  the  Organization,  Scope  and  Fi- 
nances of  the  Public  School  System  of  Oakland,  California,"  Board  of 
Education  Bulletin,  No.  8,  June,  1915. 

2  Elliff,  J.  D.:  "A  Study  of  the  Rural  Schools  of  Saline  County, 
Missouri,"  University  of  Missouri  Bulletin,  Vol.  16,  No.  22,  p.  8, 
footnote 


Translating  Statistical  Material  315 

found  it  customary  for  rural  teachers  to  have  only  three  years  of 
training  beyond  the  elementary  school,  but  in  order  for  him  to  ad- 
vance to  a  town  school  position,  he  had  to  get  an  additional  year  of 
training.     Etc.,   etc. 

This  has  been  imitated  since  in  many  cases.  Thus,  in 
the  American  School  Board  Journal,  for  August,  1917,  page 
70,  there  is  a  description  of  the  typical  Iowa  high  school 
principal,  based  on  medians.  The  author  has  had  this 
device  used  by  many  of  his  students  in  writing  up  in- 
vestigations. 

4.    The  Mode 

The  mode  may  be  translated  as  follows :  A  certain 
article  of  clothing  is  said  to  be  in  "  fashion  "  when  more 
people  wear  it  than  do  without  it.  Likewise  in  a  distribu- 
tion, the  mode  is  the  "  fashion  "  in  cases ;  more  appear 
there  than  anywhere  else. 

5.    Spread  or  Dispersion  or  Variability 

There  are  wide  differences  in  the  wealth  of  the  people 
of  this  country.  Wealth  of  individuals  varies  all  the  way 
from  Rockefeller  and  his  millions  to  the  poor  street 
beggar.  This  variation  in  statistics  is  called  the  spread 
or  dispersion.  Now  some  agitators,  if  they  had  their 
way,  would  eliminate  this  spread  by  making  all  people's 
wealth  equal. 

In  the  same  way  there  are  all  sorts  of  variations  in 
children  in  school  work,  for  any  particular  line  of  work. 
It  would  be  just  as  great  an  error  as  that  of  the  agitators 
to  reduce  the  variability  in  any  one  line  so  that  all  children 
were  considered  equal  in  performance. 

Professor  J.  F.  Bobbitt,  in  the  School  Review  for  October, 
1915,  page  508,  uses  the  translation  "  zone  of  safety  "  to 


316  School  Statistics  and  Publicity 

indicate  that  on  high  school  costs  of  instruction,  it  would 
be  well  to  try  to  get  within  the  middle  50  per  cent. 
That  is,  he  translates  the  spread  between  the  quartiles  by 
"  zone  of  safety." 

6.    Correlation 

The  Biblical  phrase,  "  the  first  shall  be  last  and  the  last 
shall  be  first,"  might  be  used  to  good  advantage  in  trans- 
lating a  perfect  negative  correlation.1 

The  following  is  a  good  translation  of  a  coefficient  of 
correlation  of  .48  between  abilities  in  shop  practice  and 
abilities  in  drawing : 

There  is  marked  evidence  that  abilities  in  shop  practice  and  drawing 
accompany  each  other.  Students  above  the  average  in  one  group  will 
tend  to  be  above  the  average  in  the  other.  It  is  not  known  specifically 
in  what  way  the  two  abilities  are  centrally  connected,  or  to  what  ex- 
tent the  presence  of  either  one  is  an  indication  of  the  other.2 

EXERCISE 

Take  the  school  report  or  school  survey  used  in  the  exercises  on 
pages  233  and  302.  Write  out  a  detailed  criticism  of  the  transla- 
tions of  statistics  or  lack  of  them  in  it  from  the  standpoint  of  their 
effectiveness  with  the  public,  showing  just  why  they  are  good  or  liable 
to  be  unsuccessful.  In  the  cases  of  the  unsuccessful  ones,  or  failure 
to  use  translations  where  desirable,  make  up  translations  that  would 
present  the  same  data  properly. 

REFERENCES    FOR    SUPPLEMENTARY    READING 

Ellis,  A.  Caswell.     "The  Money  Value  of  Education."     United  States 

Bureau  of  Education  Bulletin,  1917,  No.  22. 
McAndrew,  William.     The  Public  and  Its  School. 

'  Suggested  by  one  of  the  writer's  students,  Mr.  L.  A.  Sharp 
2  Rugg,  II.  O.  :    Statistical  Method  Applied  to  Education,  p.  257 


SELECTED   AND   ANNOTATED    BIBLIOGRAPHY 

The  aim  in  this  is  to  give  a  minimum  list  of  the  simpler  and  more 
easily  accessible  materials. 


I.  Statistical  Method 

Chapman  and   Rush.      The   Scientific  Measurement   of   Classroom 
Products.     Silver,  Burdett  and  Company,  Boston,  1917. 

Contains  excellent  brief  chapters  on  the  theory  of  scales,  their 
application  in  schools,  and  dangers  incident  to  their  use.  Other 
chapters  present  the  more  important  scales  lor  measuring  work 
in  the  formal  subjects  in  the  elementary  school,  describe  processes 
for  getting  results,  and  show  how  the  results  may  be  used  to 
better  classroom  work. 
Elderton,  W.  P.  and  E.  M.  Primer  of  Statistics.  A.  &  C.  Black, 
London,  1910. 

A  brief,  very  simple,  and  readable  treatment,  with  no  special 
reference  to  education. 
King,  W.  I.     The  Elements  of  Statistical  Method.     The  Macmillan 
Company,  New  York,   1915. 

An  elementary,  concise,  straightforward  treatment,  but  adapted 
more  to  economic  or  historical  work  than  to  school  problems. 
Monroe,   W.   S.     Educational   Tests  and  Measurements.     Riverside 
Press,  Cambridge,  Mass.,  1917. 

A  treatment  which  works  in  many  of  the  elements  of  sta- 
tistics very  simply  and  forcibly,  under  discussions  of  various  tests 
and  scales. 
Rugg,  H.  O.     Statistical  Methods  Applied  to  Education.     The  River- 
side Press,  Cambridge,  Mass.,  1917. 

An  admirable  book  for  its  general  purpose,  emphasizing  the 
problems  of  the  school  administrator.  The  statistical  part  proper, 
while  written  as  much  as  possible  in  a  non-technical  style,  is  natu- 
rally carried  to  a  much  greater  refinement  and  intricacy  than  are 

317 


318  School  Statistics  and  Publicity 

necessary  in  preparing  school  statistics  for  publicity.  However, 
the  last  chapter  is  very  fine  for  publicity  work.  The  bibli- 
ography covers  all  the  main  problems  which  superintendents  need 
to  study  quantitatively,  especially  surveys,  and  is  alone  worth 
the  price  of  the  book. 
Thorndike,  E.   L.     An  Introduction   to  the  Theory  of  Mental  and 

Social  Measurements.     Teachers  College,  Columbia  University, 

New  York,  1913. 

A  complete  exposition  of  things  needed  in  the  fields  indi- 
cated by  its  title.  It  has  few  direct  applications  for  the  admin- 
istrator, and  is  extremely  difficult  for  the  beginner  in  statistics. 

II.   Calculating  Tables 

Crelle,  A.  L.     Rechentafeln.     G.  Reiner,  Berlin,  new  edition,  1907. 

Gives  products  to  1000  by  1000. 
Peters,    J.     Neue    Rechentafeln   fur    Multiplikation    und    Division. 

G.  Reiner,  Berlin. 

III.   Exercises  and  Problems 

Thorndike's  Mental  and  Social  Measurements  and  Rugg's  Statistical 
Methods  Applied  to  Education  have  problems  in  various  places, 
some  of  which  may  be  easily  adapted  for  practice  work. 

Rugg,  H.  O.  Illustrative  Problems  in  Educational  Statistics.  Pub- 
lished by  the  author,  University  of  Chicago  Press,  1917. 

IV.   Graphic  Methods 

Brinton,  W.  C.  Graphic  Methods  of  Presenting  Facts.  Engineering 
Magazine  Company,  New  York,  1914. 

An    excellent    non-technical    treatment,  profusely    illustrated. 
While  not  written  especially  for  school  men,  its  conclusions  and 
suggestions  are  easily  adapted  to  school  problems. 
Ellis,  A.  Caswell.     The  Money  Value  of  Education.    Bulletin  of  the 
United  States  Bureau  of  Education,  1917,  No.  22,  Washington. 
Contains  numerous  charts  used  in  educational  campaigns. 
RUGG,  H.  ().     Statistical  Methods  Applied  to  Education.     The  River- 
side1 Press.  Cambridge,  .Mass..  1917. 

Chapter  X  presents  numerous  good  examples. 


Selected  Bibliography  319 

V.   School  Reports,  General  Treatments 

Bliss,  D.  C.  Methods  and  Standards  for  Local  School  Surveys.  D.  C. 
Heath  and  Company,  Boston,  1918. 

A  simple  but  adequate  treatment  of  the  topics  indicated,  which 
will  be  of  great  value  to  the  superintendent.  Has  many  good 
tabulations  and  some  graphs. 

Giles,  J.  T.  A  Statistical  Study  of  School  Reports  from  the  Twenty- 
five  Largest  Cities  of  Indiana.  Educational  Administration  and 
Supervision,  Vol.  II,  pp.  305-311. 

Hanus,  Paul  H.  School  Efficiency.  A  Constructive  Study.  School 
Efficiency  Series.  World  Book  Company,  Yonkers-on-Hudson, 
N.Y.,  1913. 

A  study  of  twenty-six  widely  selected  city  reports  in  the  United 
States. 

Snedden,  David  S.  and  Allen,  William  H.     School  Reports  and 
School  Efficiency.     The  Macmillan  Company,  New  York,  1908. 
A  pioneer  book  in  this  field,  now   useful  chiefly  for  its  sug- 
gestions on  what  to  include  in  a  report  and  on  using  tabulations. 

VI.    School  Reports  and  Surveys  Especially  Valuable  from  a 
Publicity  Standpoint 

The  publisher  is  indicated  for  each  item  given  here.     For  a  copy 
of  any  other  survey  or  report  mentioned  in  the  body  of  the  text, 
address  the  superintendent  of  the  school  system  concerned. 
Alabama.     An  Educational  Survey  of  Three  Counties  in  Alabama. 

Department  of  Education,  Montgomery,  Ala.,  1914. 
Boston,  Massachusetts.     Report  of  a  Study  of  Certain  Phases  of 

the   Public   School   System   of  Boston,    Massachusetts.     Teachers 

College,  Columbia  University,  New  York  City,  1916. 
Butte,  Montana.     Report  of  a  Survey  of  the  School  System  of  Butte, 

Montana.     By  Strayer,   G.  D.,  and  others.      Board    of   School 

Trustees,  1914. 
Cleveland,  Ohio.     The  Cleveland  Education  Survey.     Ayres,  L.  P., 

Director.     Published    in    twenty-five    separate    monographs    by 

the  Survey  Committee  of  the  Cleveland  Foundation,  Cleveland, 

Ohio.     The  following  are  especially  valuable: 

Child  Accounting  in  the  Public  Schools  —  Ayres 
Financing  the  Public  Schools  —  Clark 


320  School  Statistics  and  Publicity 

Measuring  the  Work  of  the  Public  Schools  —  Judd 
The  Cleveland  School  Survey  (summary)  —  Ayres 

Dansville,  New  York.  A  Study  —  The  Dansville  High  School. 
By  Foster,  J.  M.     F.  A.  Owen  Publishing  Co.,  Dansville,  N.  Y. 

Denver,  Colorado.  Report  of  the  School  Survey  of  School  District 
Number  One  in  the  City  and  County  of  Denver.  Part  I,  General 
Organization  and  Management;  Part  II,  The  Work  of  the  Schools; 
Part  III,  The  Industrial  Survey;  Part  IV,  The  Business  Manage- 
ment;  Part  V,  The  Building  Situation  and  Medical  Inspection. 
The  School  Survey  Committee,  Denver,  Colorado,  1916. 

Des  Moixes,  Iowa.  Annual  Report  of  the  Des  Moines  Public  Schools. 
For  the  year  ending  July  1,  1915.  Board  of  Education,  Des 
Moines,  Iowa. 

Ellis,  A.  Caswell.  The  Money  Value  of  Education.  Bulletin  of 
the  United  States  Bureau  of  Education,  1917,  No.  22. 

Grand  Rapids,  Michigan.    School  Survey.    By  a  large  staff.    1916. 

Janesville,  Wisconsin.  An  Educational  Survey.  By  Theisen, 
W.  W.,  and  staff  of  state  department.  Published  by  State  De- 
partment of  Public  Instruction,  Madison,  Wisconsin. 

McAndrew,  William.  The  Public  and  Its  School.  World  Book 
Company,  Yonkers-on-Hudson,  N.  Y.,   1916. 

Minneapolis,  Minnesota.  Three  Monographs  on  School  Finance 
in  Minneapolis.  By  Spaulding,  F.  E.  Board  of  Education, 
Minneapolis,   Minn. : 

A  Million  a  Year 

Financing  the  M inneapolis  Schools 

The  Price  of  Progress 

Nkwrurgh,  New  York.  .  The  Ncirburgh  Survey.  Department  of 
Surveys  and  Exhibits,  Russell  Sage  Foundation,  12S  East  23d 
Street,  New  York  City,  1913. 

Newton,  Massachusetts.  The  Newton  Public  Schools.  By  Spauld- 
ing, F.  E.     191.2  and  1913,  Newton,  Mass.    (Out  of  print.) 

NEW  York  City.  Report  of  Committee  on  School  Inquiry.  By  Hanus, 
Paul  H.,  and  others.  School  Efficiency  Series,  World  Book 
Company,    Yonkers-on-Hudson,  N.  Y. 

Ohio.  Report  of  the  Ohio  State  School  Survey  Commission.  By  Camp- 
bell, M.  E.,  Allendorf,  W.  L.,  and  Thatcher,  C.  J.,  1914. 

Portland,  Orkgox.  The  Portland  Surrey.  By  Cubberley,  E.  P., 
and  others,  1913. 


Selected  Bibliography  321 

Report  of  the  Committee  on    Uniform  Records  and  Reports.     U.   S. 

Bureau  of  Education  Bulletin,  1912,  No.  3. 
Rockford,  Illinois.     A  Review  of  the  Rockford  Public  Schools,  1915- 

1916.     Board  of  Education,  Rockford,  Illinois,  1916. 
Salt  Lake  City,  Utah.     Report  of  a  Survey  of  the  School  System  of 

Salt  Lake  City,  Utah.     By  Cubberley,  E.  P.,  and  others.     Board 

of  Education,  Salt  Lake  City,  1915. 
San  Antonio,  Texas.     The  San  Antonio  Public  School  System.     By 

Bobbitt,  J.  F.     The  San  Antonio  School  Board,  1915. 
Springfield,  Illinois.     The  Public  Schools  of  Springfield,  Illinois. 

By  Ayres,  L.  P.     Division  of  Education,  Russell  Sage  Founda- 
tion, Naw  York  City,  1914. 
St.  Louis,  Missouri.     Report  of  Survey  of  St.  Louis  School  System. 

Board  of  Education,  1917. 
Texas.     A  Study  of  Rural  Schools  in  Texas.     By  White,  E.  V.  and 

Davis,  E.  E.     University  of  Texas,  Austin,  Texas,  1914. 
United  States.     A  Comparative  Study  of  Public  School  Systems  in 

the    Forty-eight    States.      Division    of    Education,    Russell    Sage 

Foundation,  New  York  City,  1912. 

This  is  now  out  of  print,  but  copies  will  doubtless  be  available 

for  the  superintendent  at  his  state  department  of  education  or 

state  university.     Copies  were  sent  to  all  the  members  of  the 

state  legislatures  at  the  time  of  its  issue,  1912. 


INDEX 

Note.  —  To  save  space,  the  words  education,  publicity,  school, 
statistics,  and  teachers  are  in  the  main  omitted  from  this  index.  In 
the  numerous  combinations  in  which  they  naturally  occur,  look  for 
the  next  most  significant  word. 


Absence,  translation  for,  307 

Absurdities  in  rank-order  combi- 
nations, 198 

Accuracy,  187-190,  230;  errors 
in  attempts  at  too  great  ac- 
curacy, 16-19  ;  in  graphs,  283  ; 
in  translations,  308 

Advertising,  school,  error  in  bas- 
ing upon  exceptional  graduates, 
23 

Age-grade  tables,  emphasis  in,  228 

Age-progress,  table  form  for,  229 ; 
circle  graphs  for,  251 

Age,  school,  error  in  determining, 
5 

Ages  of  pupils,  distribution  table 
for,  120  ;     graphs  for,  121 

Aikins,  Professor,  47 

Alabama,  Survey  of  Three  Counties, 
239,  250,  288,  289,  290 

Allen,  W.  H.,  25,  27,  33,  72,  78, 
201,  202,  213,  216 

Alphabetical  order  in  tables,  221- 
222 

Alternate  columns  in  tables,  218 

Amarillo,  Tex.,  68 


American  Book  Company,  graph, 
241 

Area  graph  for  comparison  on 
component  parts,  245 ;  for 
comparisons  with  circles,  251 

Arithmetic  tests,  46,  47,  49 ;  sur- 
face of  frequency  for,  117; 
frequency  table  for,  134 ;  table 
for  results,  232  ;  variability  in, 
171 

Arkadelphia,  Ark.,  8 

Artistic  features  in  tables,  220 

Association  of  science  and  mathe- 
matics teachers,  table  to  show 
growth,  232 

Atlanta,  300,  301 

Attendance,  at  teachers'  associa- 
tions, error  in  computing,  14 ; 
school,  blanks  for  showing,  72, 
73 ;  error  in  indefinite  units, 
4 ;  perpetual  graph  device  for 
showing,  298  ;     problems  of,  34 

Automobile  graph,  290,  296 

Average,  141-147 ;  advantages 
of,  145;  computation  of,  long 
method,    141-142 ;      computa- 


323 


324 


Index 


tion  of,  short  method,  142 ; 
definition  of,  141 ;  disadvan- 
tages of,  146 ;  errors  in  com- 
puting, 19-21 ;  graphic  repre- 
sentation of,  144 ;  translation 
for,  313 

Average  deviation,  156 

Averages,  deviations  from,  21-22 

Ayres,  L.  P.,  5,  166,  241,  243, 
244,  251,  259,  291,  292,  310 

Ayres  handwriting  scale,  53 

Ayres  spelling  scale,  106 

Background  for  graphs,  241,  283 

Bagehot,  Walter,  90 

Baltimore  Survey,  248 

Bar  graph,  237 ;  cartoon  effect, 
239,  260,  288;  comparisons 
with,  243,  244,  253;  com- 
ponent parts,  use  for,  239 ; 
curve  effects  with,  263 ;  order 
of  items  for,  241;  right  and 
left  form,  247,  248 

Bi-modal  distribution,  147 

Bird's-eye  view  through  tabula- 
tion, 209 

Birmingham,  Ala.,  210,  298 

Black,  use  of,  on  maps,  293 

Blanks,  71-78;  making,'  78; 
examples  of  good,  78 ;  re- 
vision of,  42 ;  vs.  card  index, 
79-81 

Bloomington,  Ind.,  172 

Bobbitt,  J.  F.,  17,  22,  69,  70,  86, 
95,  96,  101,  162,  167,  315 

Bobbitt  table,  18,  96,  103,  104, 
127,  130,  147,  152,  153,  156, 
161,  162,  165,  169,  223,  225, 
226,  240,  293,  295 


Bold-face  type,  use  in  tables,  219, 
227 

Boston,  circulation  of  school  re- 
port, 27 

Boston  Report,  245,  246,  269 

Bowley,  A.  L.,  189 

Bridgeport  Survey,  229,  255,  257, 
258 

Brief,  use  of,  in  planning,  42 

Brinton,  W.  C,  116,  242,  272, 
276,  286;  graphic  methods  of 
presentation,  242 ;  rules  for 
graphic  presentation,  272-275 

Buffalo,  213 

Buildings,  lack  of,  cartoon  effect 
for,  301 ;  problems  on,  34 ; 
table  for,  213;  units  and 
scales  for,  53 

Butte  Survey,  117,  128,  130,  167, 
293 

By-products  in  collecting  data, 
41 

Calculating  devices,  191 
Calculating  tables,  191 
Calculation,    economies    in,    190- 

192 
Card  index  vs.  one  blank,  79-81 
Carelessness  in  securing  data,  15 
Cartoon    effects    in    graphs,    256, 

287,  301 
Cartoons,   economies   in  making, 

296 
Census,  school,  problems  of,  34 
Central    tendency,    measures    of, 

124-147 
Charts,  time,  261 
Checking,    in    calculations,    190 ; 

on  blanks,  84 


Index 


325 


Chicago,  University  of,  statistical 
method  in  School  of  Education, 
29;    blank  for  rating  teachers, 
270 
Cincinnati,  214 

Circle  graph,  235,  259 ;    for  com- 
parisons,   250,    254,    255,    256, 
259 ;     with  cartoon  effect,  258, 
259 
Cleveland,  5,  36,  174,  214 
Cleveland      Survey,      118,      154, 
159,   251,   253,   256,   282,    294, 
313 
Coefficient     of     correlation,     cal- 
culation   of,     184,     185;      ex- 
amples of,   183 ;     meaning  of, 
181,  182 
Coefficient  of  variability,  170 
Coffman,  L.  D.,  314 
Collecting  data  with  high  school 
-     students,  86-88 
College    degrees,    error    in    com- 
parisons with,  8 
College  salaries,  error  in  getting 

average,  19 
Columbus  Dispatch,  290 
Committee     on     standards     for 

graphic  presentation,  267 
Committee    on    uniform    records 

and  reports,  58 
Comparisons,  simple,  240  (see 
also  Relationships,  164-186) ; 
using  component  parts,  14,  266, 
267;  using  percentages,  13, 
14;  using  indefinite  units, 
3-16  ;  using  relative  position, 
269 ;  using  unsound  treat- 
ment, 3-16  ;  with  bar  graphs, 
239,   240,   243,   244,   253,   266, 


267,  269;  with  circle  graphs, 
250,  256,  258;  with  cartoon 
effects,  256,  258;  with  curves, 
104,  262,  263,  266,  267;  with 
triangle  graphs,  257 

Component  part  graphs,  236, 
238,  239,  242,  245,  254,  255 

Composition  scales,  46,  53 

Composition  tests,  frequency 
table  for,  131 ;     graph  for,  168 

Concentric  circle  graph,  252 

Constant  errors,  188 

Contests,  judging,  11-12,  52, 
192  ff. 

Continuous  series,  48  ;  graph  for, 
165 

Conway,  Ark.,  Survey,  314 

Correlation,  173  (see  also  Co- 
efficient of  Correlation) ; 
graphic  devices  for  showing, 
175-181 ;  like-signs,  table  for, 
178-179  ;    translation  for,  316 

Cost  of  instruction,  16,  17,  18, 
48,  55,  70,  96,  101,  104,  105, 
130,  150,  222,  223,  244,  307, 
308 

Cost  of  maintenance,  218,  264 

Cost  per  pupil,  263 

Cost  records,  cartoon  effect  to 
show  value  of,  287 

Costs,  cartoon  for,  290 ;  errors 
in  computing,  16;  graphs  for, 
288,  292;  sampling  for,  68; 
translation  for,  307,  308 ;  units 
and  scales  for,  55. 

Courtis  arithmetic  tests,  46,  47, 
142,  143,  149,  199;  frequency 
table  for,  134  ;  graphs  for,  172, 
266 


326 


Index 


Crayons  for  graphs,  296 

Crelle  tables,  use  of,  191 

Cross-section  paper,  for  computa- 
tion, 190;    for  large  graphs,  296 

Cubberley,  E.  P.,  263,  265,  314 

Current  school  reports,  200 

Curve  effects  with  bar  graphs, 
263,  266 

Curves,  arbitrary  signs  for,  166; 
for  comparisons,  262  ;  for  dis- 
crete scales,  104 ;  standards 
for,  266-268 

Data,  carelessness  in  securing, 
15;  collection  of ,  33-89  ;  econ- 
omies in  collecting,  82-88 ; 
sources  of,  58-61 

Dearborn,  W.  F.,  106,  107,  108, 
180 

De  Voss,  J.  C,  271 

Defectives,  unanalyzed  total  for, 
3  ;  number  of,  translation  for, 
309 

Degrees,  college  and  university, 
error  in  comparisons  with  8 

Des  Moines,  207,  210,  219,  257 

Detroit,  217 

Deviation,  coefficient  of,  170 ; 
measures  of,  149-163 ;  graphic 
representation  of,  101,  137; 
measure  for  given  distribution, 
162;  translation  for,  315 

Dimmitt,  R.  L.,  298 

Discrete  scale  or  series,  definition, 
48  ;  graphic  representation  of, 
103-105,  164 

Dispersion  (see  Deviation) 

Distance,  map  to  show  by  time 
elements,  280 


Distribution,  of  cases  on  a  map, 
277  ;  of  pupils,  age-grade  blank 
for,  73 ;  of  school  moneys, 
error  in,  5 ;  of  time  in  ele- 
mentary grades,  graph  for,  236  ; 
tables,  106-111;   graphing,  111 

Dollar  circle  graphs  for  component 
parts,  254 

Dollar  graph  for  proportionate 
parts  with  cartoon  effects,  258, 
259 

Dollar  proportionate  parts  table, 
230,  231 

Dollar-sign  graph,  260,  289 

Dots  for  bar  graphs,  292 

Double  distribution  table,  72 

Double  entry  table  for  receipts 
and  payments,  211 

Economies,  in  collecting  data,  82- 
88  ;  in  graphs,  296 

Education,  value  of,  graph  to 
show,  247  ;    translation  for,  311 

Educational  investigations,  knowl- 
edge needed  for,  94 

Efficiency  record  of  teachers, 
summarizing  graph  for,  720 

Elimination,  bar  graph  for,  243; 
of  high  school  students,  table 
for,  217  ;     problems  of,  34 

Elliff,  J.  D.,  314 

Emphasis  in  tables,  227 

Enrollment,  errors  in,  3-5 ; 
graphs,  248,  249,  252,  253,  261, 
294 ;  map  devices  to  show 
distribution,  277,  278  ;  N.  E.  A. 
blank  for,  74 ;  translation  for, 
310;  units  and  scales  for,  57 

Errors,  constant,  188 ;  variable,  187 


Index 


327 


Estimating  reliability,  189 
Exaggerations  in  graphs,  283 
Expenditure    graphs,     245,     259, 

265,  269 

Expenditures,  average,  62  ;  errors 
in  comparisons,  4 ;  problems 
of,  34 ;  proportionate  units 
and  scales  for,  54 ;  sampling 
for,  68,  70;  tables,  214,  215; 
translation  for,  308  (see  also 
Costs) 
Extreme  range  variation,  149 
Eyestrain,  avoiding,  218,  227 

Fisher,  Irving,  242 

Florida,  20 

Ford  automobile  translation,  306 

Fractions,    errors   in   getting   too 

small,  17 
Frederic,  Wis.,  88 
Frequency,  surface  of,   111,   116, 

123  ;     multi-modal   surface   of, 

125;     normal   surface  of,   117; 

skew     surface      of,      118-122 ; 

tables,  106-111 

Georgia,  University  of,  279 

Giles,  J.  T.,  200 

Grading   pupils,    10,    11,    21,    22, 

63,    66,    107,    108,    170,    196; 

blank    for    studying    teachers' 

standards,  76 
Graphs,  234-301  ;     check  list  for, 

274 ;       economies    in    making, 

296  ;      examples   of  good,   288  ; 

size    of,    282 ;      standards    for, 

266,  272,    281;      summarizing, 
268 

Gray,  W.  S.,  48 


Grounds,  size  of,   cartoon  graph 

for,  260;    translation  for,  310 
Grouping,  41,  107-110,  210 
Gummed  letters  for  graphs  and 
charts,  297 

Haggerty,  M.  E.,  171,  172,  176, 
199 

Hammond  Survey,  220 

Handwriting  scales',  45,  50,  53 

Handwriting  tests,  distribution 
table  for,  128 ;  surface  of  fre- 
quency at  Cleveland,  118 

Hanus,  Paul,  25,  27,  200,  201 

Harvard-Newton  composition 

scale,  53 

Headings,  for  graphs,  281 ;  for 
tables,  209,  219;  form  for 
printing,  216 

Health  education,  lack  of,  trans- 
lation for,  311 

Heating,  22 

Herrick,  W.,  26 

High  school  enrollment,  graph 
for,  293 

High  school  training,  value  of, 
graph  to  show,  247 

Hillegas  composition  scale,  53, 
131 

Histogram  or  column  diagram, 
111-116,  122  ;  check  form,  114, 
115 

Illiteracy,  cartoon  graph  to  show, 
288  ;  error  in  treatment  of,  13, 
16 

India  ink  for  graphs  and  charts, 
297 

Indiana,  199,  200 


328 


Index 


International  Harvester  Company 
graph,  247 

Jingle  fallacy,  47 
Judging    contests,     11,     12,     52, 
192  ff. 

Kelley,  F.  J.,  184,  271 

Key    numbers,    for    blanks,    75 ; 

for  tables,  219 
King,  W.  I.,  36,  38,  39,  41,  91, 

92,  95,  97,  145,  313 
Kirk,  Jno.  R.,  65 

Length  of  school  year,  bar  graph 
for,  291 ;     translation  for,  307 

Lettering  on  graphs,  296 

Library  Bureau,  59 

Library  statistics,  error  in  in- 
definite units,  10 

Lighting  space,  graphs  for,  286, 
295 

Lines,  clear  in  graphs,  297 ; 
dividing,  in  tables,  218,  219 

Louisville,  6,  231,  244,  258,  283 

Lowry,  F.  C,  289 

MacAndrew,  Wm.,  300,  304 

Maclin,  E.  S.,  300 

Maintenance    cost,    per    cent    of, 

table  for,  218 
Maps,  275  ff . ;     use  of  black  on, 

293 
Marking  pupils,  10,  11,  21,  22,  63, 

66,  107,  108,  170,  196;     blank 

for  studying  teachers'  standards 

on,  76 
Masonic  pin  translation,  306 
Maxwell,  Wm.,  25,  26,  37 


Median,  advantages  of,  138 ; 
computation  of,  129-137 ;  def- 
inition of,  128 ;  disadvantages 
of,  139 ;  graphic  representa- 
tion of,  137;  translation  of,  313 

Median  deviation,  155 

Medical  inspection,  unanalyzed 
totals  in,  3 ;     problems  in,  35 

Membership,  table  for  showing 
growth  in,  232 

Memphis,  166,  265 

Meyer,  Max,  11,  63 

Minneapolis,  63,  70,  145 

Minnesota,  9 

Missouri,  University  of,  grading 
system,  11;  State  Department 
of  Education,  279 

Mobile,  27 

Mode,  advantages  of,  126 ;  def- 
inition, 124 ;  computation  of, 
125 ;  disadvantages  of,  127  ; 
graphic  representation  of,  125 ; 
translation  of,  315 

Monroe,  W.  S.,  136,  271 

Monument  graph,  249 

Multimodal  surface  of  frequency, 
125 

Nashville,    Tenn.,    77,    119-121, 

280 
National   Education   Association, 

58,  74,  78 
Neatness  in  tables,  220 
Negroes,  schooling  for,  15,  307 
Newburgh  Survey,  259,  260,  261 
Newton,     Mass.,    40,     102,     103, 

261,  263,  288,  300 
New  York  Survey,  27,  303 
Normal  schools,  2,7,  150 


Index 


329 


Normal  surface  of  frequency,  117- 
119 

Oakland  Survey,  218 

Ohio  Survey,  259 

Old  Man  Ohio  cartoon,  290 

Omitting  important  factors,  15 

Order  of  items  for  bar  graph,  241 ; 

for  tables,  221,  223 
Over-age,    error    in    method    of 

determining,  5 

Paper     letters     for     charts     and 

graphs,  297 
Peabody     (George)     College     for 

Teachers,  277 
Pearson  coefficient  of  correlation, 

158,  185 
P.  E.,  or  probable  error,  155 
Penmanship  (see  Handwriting) 
Percentage  tables,  graphing,  167 
Percentages,  errors  in,  13-15 
Percentile  deviations,  153 
Percentiles,  153 

Perpetual   attendance   graph   de- 
vice, 298 
Phelps,  S.  J.,  86 
Phi  Beta  Kappa,  11 
Planning  statistical  treatment,  38- 

43 
Playgrounds,   cartoon   graph   for, 

260;      size   of,   translation   for, 

310)    units  and  scales  for,  53 
Population     of     school     district, 

units  and  scales  for,  56 
Portland  Survey,  51,  52,  224,  227, 

230,  282 
Presentation    of   school   statistics 

to  public,  errors  in,  24 


Probability  surface  of  frequency, 

118 
Problem,  statistical,  how  to  state, 

39 
Property,    value    of,    translation 

for,  309  ;  value  of,  graph  for,  289 
Proportionate  parts  table,  231 
Public  indifference,  26-27 
Puckett,  W.  F.,  5 

Q  or  Quartile  Deviation,  151,  153  ; 
graphic  representation,  137 ; 
translation  for,  315 

Quartiles,  151;  graphic  repre- 
sentation, 137 

Questionnaire  method,  value  of, 
59  ;  blank  for,  65  ;  sampling 
for,  66 

Range,  measures  of,  149-163 
Rank-order  combinations  of  data, 

192-198 
Rating  of  teachers,  summarizing 

graph  for,  270 
Receipts  and  payments,  problems 

of,  34;  table,  211 
Records,  58,  287 
Red  Cross,  279 
Relationships,  164-186 
Relative  position,  bar  graph  for, 

269 
Reliability,  187-190 
Reports,  25-27,  40 
Reproducing  graphs  for  the  public, 

297 
Retardation,  error  in  determining, 

6;     graph  for,   166,  237,  257; 

problems    of,    35 ;      translation 

for,  306,  310 


330 


Index 


Revenues,  errors  in  comparison, 
15 

Rice,  Dr.,  9,  45 

Rockford  Review,  231,  236,  252, 
264,  279,  285 

Rubber  stamps  for  graphs  and 
charts,  297 

Rugg,  H.  O.,  67,  79,  182,  316 

Ruled  blank  book,  77 

Ruler  strip  device  for  use  on  re- 
ports, 85,  86 

Rural  school  work,  map  for,  279 

Russell  Sage  Foundation,  166, 
243,  244,  257,  259,  260,  261 

Salaries,  janitors',  257  ;  principals', 
257;  teachers',  22,  109,  110, 
125,  150,  154,  290;  teachers', 
error  in  indefinite  units,  6,  7 ; 
teachers',  units  and  scales  for, 
54 

Salt  Lake  City  Survey,  122,  125, 
168,  219,  232,  237,  2S2,  293 

Sampling,  22-24,  62-71 

San  Antonio  Survey,  17,  22,  86, 
167,  225 

San  Francisco  Survey,  66 

Scales,  24,  43-53,   100;     discrete 
and  continuous,  48;     example 
of,    53-57  :      graphic    represe.  - 
tation  of,  101,  285;     objective, 
44 ;  subjective,  44 

Schedules,  teachers',  error  in  com- 
puting with  indefinite  units,  7 

Schooling,    value    of,    translation 
',  311 

S  'hool  problems,  34-  37 

School  statistics,  errors  in,  1-24: 
errors    in    presentation    of,    24: 


need   for  better,    1-32 ;      value 

of,  for  superintendents,  30-31 
Scoring  data,  economies  in,  82-84 
Semi-inter-quartile  range,  151 
Sequence  in  tables,  221 
Seymour,  F.  O.,  68 
Sharp,  L.  A.,  316 
Shaw-Walker  Company,  59 
a  =  Greek       sigma  =  abbreviation 

for  Standard  Deviation 
Signs,      arbitrary,      for      graphs, 

299 
Skew     distribution,     148;  devia- 
tions for,  159 
Skewness,  118-122 
Skew  surface  of  frequency,   118- 

122 
Smoothed   surface    of    frequency, 

119 
Smoothing  graphs,  113 
Snedden,   David,  25,   27,   33,   72, 

78,  201,  202,  213,  216 
South  Bend  Survey,  209,  221 
Spaulding,  F.  E„  40,  43,  63,  70, 

71,  103,  145 
Special  classes,  35 
Spelling,  errors  in   lack   of   units 

for,    9;  scale,    106;      tests,    45, 

122,  167,  169 
'•'•-read,     measures    of,     149   163; 

translation  for,  315 
Springfield  Survey,  211,  243,  257, 

313 
Standard  Deviation,  158 
Standard  tests,  sampling  in,  66 
Standards,  errors  in  striving  for, 

17-19 
Statistical    method,    reliability   of 

results    with.     187,     189,    190: 


Index 


331 


value  of,  and  when  to  use,  19- 
24,  28,  29,  30,  33,  36,  37,  38- 
40,  91-93,  95,  97-99,  100,  200 

Step  method,  economy  for  com- 
putation, 191 

Step  on  a  scale,  meaning  of,  24,  49 

Stone  tests,  117 

Strayer,  G.  D.,  62,  79,  119 

Street  maintenance,  table  for, 
225 

Students,  use  of,  for  statistical 
work,  192,  299 

Subjective  scales,  46 

Summarizing  data  on  blanks,  75, 
76 

Summary  tables,  221 

Summer  session  enrollment,  error 
in  indefinite  units,  3 

Supervision,  problems  of,  35 

Surface  of  frequency,  111-122 

Symmetrical  distribution,  117 

Tables,     distribution,     106 ;       of 

frequency,  106  ;    series  of,  220 
Tabulation,    209,    233;      for    the 

public,  203,  204,  206-233 
Taxes,  increase  in,  bar  graph  for, 

261 ;      increase    in,    translation 

for,  311 
Tax   rate,    errors    in    comparison 

with,    7,    8;      graphs   for,    102, 

265 ;    table  for,  224  ;    units  and 

scales  for,  56,  57 
Teachers  College,  use  of  statistical 

method  at,  28 
Teaching  staff,   units   and  scales 

for,  54 
Technical     methods     needed     in 

school  statistics,  19-24,  90  99 


Tests,    rank-order    combinations 

for,  192  ;     summarizing  graphs 

for,  168,  271 
Textbooks,  graph  for  cost  of,  242 
Thermometer  graph,  102,  288 
Thorndike,  E.  L.,  30,  45,  46,  47, 

48,  49,  79,   92,   109,   144,   189, 

191,  206 
Thorndike  handwriting  scale,  45, 

53 
Tie  rankings,  195 
Title    of    graph    or    chart     (see 

Headings) 
Time  charts,  261 
Time  spent  on  each  subject,  units 

and  scales  for,  56 
Time  unit  for  translations,  306 
Totals,    forms    for    emphasizing, 

216  ;    unanalyzed,  error  in,  2-3 
Training  of  teachers,  graphs  for, 

256,  259  ;    units  and  scales  for, 

54 
Translation   of  statistics   for  the 

public,  303-316 
Triangle  graphs,  254 
Truancy,  problems  of,  35 
Two-way  tables,  72 
Type,  measures  of,  124-147 

Unclassified  items  table,  214,  215 
Unequal    things,    errors    in    con- 
sidering equal,  9-10,  47 
Uniform  records  and  reports,  233 
Units,   errors   in,   3-12,    16;     ex- 
amples of,  53-57 ;     how  to  de- 
termine, 43-57 
Updegraff,  Harlan,  70 
U.   S.   Bureau  of  Education,   59, 
120,  247,  248,  311 


332 


Index 


Uselessness  of  statistics  in  school 
reports,  200 

Valedictorian,  determining,  192 
Valid  scale,  47-48 
Vanderbilt  University,  279 
Variability,     of    children,     graph 

for,   293;      coefficient   of,    170; 

translation  for,  315 
Variable  errors,  187 
Variation,  measures  of,  149-163 
Variations,    errors    in    neglecting, 

21-22 ;      in    size    of    type    for 

tables,  232 
Variety  in  graphs,  286 
Virginia,  9,  286 

Waste,  in  school  statistics,  24- 
26;  in  teaching,  translation 
for,  311 

Wax  crayons  for  graphs  and 
charts,  296 


Wealth,  units  and  scales  for,  56 ; 

real  and  assessed  behind  each 

$1  spent  on  schools,  table  for, 

51,  52  ;    real,  table  for,  223 
Webb,  H.  A.,  306 
Weighing  machine  cartoon,  289 
Weighting   factors   in   rank-order 

combinations,  197 
Wisconsin,    Experiment    Station, 

287;       State     Department     of 

Public  Instruction,  88 
Withdrawals,  error  in  determining 

age,  6 
Womack,  J.  P.,  314 

Y.  M.  C.  A.,  280,  308,  309 

Zero   line   in    graphs,    263,    283, 

284 
Zero  point  on  scale,  46,  47 
Zeros,  use  in  tables,  216,  217 
Zone  of  safety,  18,  315 


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